The inverse scattering transform for the focusing nonlinear Schrödinger equation with
asymmetric boundary conditions
F. Demontis, B. Prinari, C. van der Mee, and F. Vitale
Citation: Journal of Mathematical Physics 55, 101505 (2014); doi: 10.1063/1.4898768
View online: http://dx.doi.org/10.1063/1.4898768
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JOURNAL OF MATHEMATICAL PHYSICS 55, 101505 (2014)
The inverse scattering transform for the focusing
nonlinear Schrödinger equation with asymmetric
boundary conditions
F. Demontis,1 B. Prinari,2,3 C. van der Mee,1 and F. Vitale3
1
Dipartimento di Matematica, Univ di Cagliari, Viale Merello 92, 09121 Cagliari, Italy
Department of Mathematics, University of Colorado Colorado Springs,
Colorado 80918, USA
3
Dipartimento di Matematica e Fisica “E. De Giorgi,” Univ del Salento and Sezione INFN,
Lecce, Italy
2
(Received 6 February 2014; accepted 2 October 2014; published online 29 October 2014)
The inverse scattering transform (IST) as a tool to solve the initial-value problem
for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary val2
ues ql/r (t) ≡ Al/r e−2i Al/r t+iθl/r as x → ∓∞ is presented in the fully asymmetric
case for both asymptotic amplitudes and phases, i.e., with Al = Ar and θl = θr .
The
is shown to be well-defined for NLS solutions q(x, t) such that
direct problem
q(x, t) − ql/r (t) ∈ L 1,1 (R∓ ) with respect to x for all t ≥ 0, and the corresponding
analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral
equations, and
as a Riemann-Hilbert problem on a single sheet of the scattering vari2
ables λl/r = k 2 + Al/r
, where k is the usual complex scattering parameter in the
IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and
transmission coefficients have a nontrivial (although explicit) time dependence. The
results presented in this paper will be instrumental for the investigation of the longtime asymptotic behavior of fairly general NLS solutions with nontrivial boundary
conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.
C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4898768]
I. INTRODUCTION
Nonlinear Schrödinger (NLS) systems have attracted the attention of the physical community
for almost 60 years, and equations of NLS type have been derived in such diverse fields as deep
water waves,4, 37 plasma physics,33 nonlinear fiber optics,21 magnetic spin waves,16, 38 Bose-Einstein
condensates,29 and much more. As a matter of fact, most dispersive energy preserving systems give
rise, in appropriate limits, to the scalar NLS, which explains the keen interest in it as a prototypical
integrable system, and motivates the efforts devoted to advance our mathematical understanding
of this equation. In this respect, the inverse scattering transform (IST) as a method to solve the
initial-value problem for the scalar NLS equation,
iqt = qx x − 2σ |q|2 q
(1.1)
(subscripts x and t denote partial differentiation throughout) has been extensively studied in the
literature, both in the focusing (σ = −1) and in the defocusing (σ = 1) dispersion regimes; see, for
instance, Refs. 2–4, 15, 26, and 37 for detailed accounts of the IST in the case of potentials q(x, t)
rapidly decaying as x → ±∞. The situation is quite different when one is interested in potentials that
do not decay at space infinity. In fact, even though the IST for the focusing NLS with rapidly decaying
potentials was first proposed more than 40 years ago, and has been subsequently the subject of a vast
0022-2488/2014/55(10)/101505/40/$30.00
55, 101505-1
C 2014 AIP Publishing LLC
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amount of studies and applications, not as much is available in the literature in the case of nontrivial
boundary conditions. The reason for this deficiency is twofold: on one hand, the technical difficulties
resulting from the non-zero boundary conditions (NZBCs) significantly complicate the formulation
of the IST; on the other hand, the onset of modulational instability, also known as the Benjamin-Feir
instability10, 11 in the context of water waves, was believed to be an obstacle to the development of
the IST, or at least to its validity. Nonetheless, a large number of exact solutions to the focusing NLS
equation with NZBCs have been found over the years by the use of direct methods. Historically, the
first such solution was found by Peregrine in 1983,30 although a solitonic solution in the presence of
a condensate was already reported in Refs. 23 and 24. In 1985, a second order Peregrine solution,
periodic in space and homoclinic in time, was found.6 “Multi-Peregrine” solutions and more general
solitonic solutions were subsequently discovered in Refs. 7–9, 22, 25, 32, and 34. In recent years
these solutions have been actively studied worldwide, and the renewed interest is due to the fact that
the development of modulation instability in the governing equation has been recently suggested as a
mechanism for the formation of “extreme” (also known as “rogue,” or “freak”) waves, where energy
density exceeds the mean level by an order of magnitude.28, 35, 36 General high-order rogue waves
in the nonlinear Schrödinger equation have also been recently obtained by the bilinear method in
Ref. 27, where it is shown that the general N th order rogue wave contains N − 1 free irreducible
complex parameters, and that the specific rogue waves obtained by Akhmediev et al. in Ref. 8
correspond to special choices of these free parameters, yielding the highest peak amplitudes among
all rogue waves of the same order. At the same time, the observation of rogue waves has been
reported in an optical system, based on a microstructured optical fiber.31 The generation of these
rogue waves has been modelled using a generalized NLS equation, and shown to be an infrequent
evolution from initially smooth pulses owing to power transfer seeded by a small noise perturbation.
In view of these recent developments, it is natural to wonder about the role that soliton solutions
play in the nonlinear development of the modulation instability, which makes the study of the longtime asymptotics of NLS solutions of great practical importance, crucial for developing a consistent
theory for rogue waves in the ocean, and for extreme events in optical fibers. In this respect, the
investigation of the IST for the focusing case with NZBCs [Eq. (1.1) with σ = −1], i.e.,
iqt = qx x + 2|q|2 q ,
(1.2)
as a means to provide the time evolution of a fairly general initial one-dimensional pulse/wave
profile over a nontrivial background, should receive a greater deal of attention, since it allows
the study of the long-time asymptotic behavior via the nonlinear steepest descent method,13, 14, 17
matched asymptotic expansions,1, 5 or other germane techniques. Until this year, the only general
study of IST for the focusing NLS with NZBCs available in the literature was in Ref. 24, which
only contains partial results as it is limited to the case of completely symmetric boundary conditions
with limx→+∞ q(x, t) = limx→−∞ q(x, t), i.e., only the case in which the potential exhibits no
asymptotic phase difference and no amplitude difference is treated. Some interesting results can
also be found in Ref. 20, where the authors propose a perturbation theory for the focusing NLS
equation with symmetric boundary conditions based on the IST in order to analyze the effect of
different dispersive, diffusive (damping), or nonlinear perturbations to the soliton propagation. Most
recently, Biondini and Kovačić12 have developed the IST for potentials with an arbitrary asymptotic
phase difference, although assuming equal amplitudes at both space infinities. They also discuss the
general behavior of the soliton solutions, as well as the reductions to all special solutions previously
known in the literature and mentioned above.
In this work, we will develop the IST for the scalar focusing NLS (1.2) with fully asymmetric
NZBCs,
q(x, t) → ql/r (t) = Al/r e−2i Al/r t+iθl/r
2
as x → ∓∞,
(1.3)
where Ar ≥ Al > 0 and 0 ≤ θl/r < 2π are arbitrary constants. This is a highly nontrivial generalization of the work,12 and it involves dealing with additional technical difficulties, the most important
of which being the fact that when the amplitudes of the NLS solutions as x → ±∞ are different,
in the spectral domain one cannot introduce a uniformization variable that allows mapping the
multiply sheeted Riemann surface for the scattering parameter to a single complex plane. From the
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point of view of physical applications, such a generalization would be particularly significant for
the theoretical investigation of rogue waves and perturbed soliton solutions in microstructured fiber
optical systems with different background amplitudes enforced at either end of the fiber. This work
would also be relevant in clarifying the role that soliton solutions play in the nonlinear development
of modulation instability in such systems.
The plan of the paper is outlined below. Section II is devoted to the study of the direct scattering
problem.
We will prove that the direct problem is well defined for potentials q(x, t) such that
q(x, t) − ql/r (t) ∈ L 1,1 (R∓ ) with respect to x for all t ≥ 0, L 1,s (R) being the complex Banach
space of all measurable functions f (x) for which (1 + |x|)s f (x) is integrable. We will then establish
analyticity of eigenfunctions and scattering data, and obtain integral representations for the latter
for potentials in this class. In Sec. III, we will formulate the inverse problem both in terms of
(left and right) Marchenko integral equations (Sec. III B), and as a Riemann-Hilbert (RH) problem
2
on a single sheet of the scattering variables λl/r = k 2 + Al/r
, where k is the usual complex
scattering parameter in the IST (Sec. III C). Important differences with respect to the symmetric
case also arise in the inverse problem, where, in addition to solitons (corresponding to the discrete
eigenvalues of the scattering problem), and to radiation (corresponding to the continuous spectrum
of the scattering operator, and represented in the inverse problem by the reflection coefficients
for k ∈ R ∪ [−i Al , i Al ]), one also has a nontrivial contribution from additional spectral data for
k ∈ (−i Ar , −i Al ) ∪ (i Al , i Ar ), which appears in both formulations of the inverse problem. [Note
that with a slight abuse of notation, in the paper we will denote the relevant segments on the imaginary
k-axis by r/l ≡ [−i Ar/l , i Ar/l ].] In particular, this implies that no pure soliton solutions exist, and
solitons are always accompanied by a radiative contribution of some sort. As a consequence, unlike
the equal-amplitude case dealt with in Ref. 12, here no explicit solution can be obtained by simply
reducing the inverse problem to a set of algebraic equations. In view of this, the present study provides
a very powerful tool for the asymptotic investigation of NLS solutions that cannot be obtained by
direct methods. Specifically, the RH formulation of the inverse problem makes it amenable to the
study of the long-time asymptotic behavior via the nonlinear steepest descent method, as was done,
for instance, in Ref. 17 for the modified KdV equation, or in Refs. 14 and 13 for the focusing NLS
with special step-like initial conditions. The Marchenko integral equations provide an alternative
setup for the study of the long-time behavior of the solutions by means of matched asymptotics, as
was recently done for KdV in Ref. 1. Section IV deals with the time evolution of eigenfunctions and
scattering coefficients, and Sec. V is devoted to some concluding remarks. For a better readability
of the paper, more technical proofs are collected in the Appendix.
II. DIRECT PROBLEM
It is well-known that the focusing NLS Eq. (1.2) can be associated with the following Lax pair:
∂v
= (−ikσ3 + Q) v ,
∂x
∂v = (2ik 2 − i|q|2 + i Q x )σ3 − 2k Q v ,
∂t
where v(x, k, t) is a two component vector, k ∈ C is the scattering parameter, and
0 −i
1 0
0
q(x, t)
σ2 =
.
,
σ3 =
,
Q(x, t) =
0
i 0
0 −1
−q ∗ (x, t)
(2.1)
(2.2)
[Here and in the following the asterisk indicates complex conjugate; σ2 is given for future reference.]
The first equation in the Lax pair is the well-known Zakharov-Shabat (ZS) scattering problem,2, 37
and the matrix Q(x, t) [or, equivalently, its entry q(x, t)] is referred to as the “potential” of the
ZS scattering problem. Here we will consider potentials with nontrivial boundary conditions (BCs)
as in (1.3), where we assume that Ar ≥ Al > 0. Note that while the asymptotic amplitudes of
the BCs Al/r can be assumed to be time independent, the asymptotic phases evolve as follows:
2
t + θl/r (see Sec. IV for details). As a result, unlike the equal-amplitude BCs
θl/r (t) = −2Al/r
(Al = Ar ) considered in Ref. 12, here it is not possible to subtract out the background and make
both BCs time-independent.
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In the formulation of the direct problem we will omit to explicitly specify the time-dependence
for brevity. It will be clear from the context whether one is considering t = 0 or an arbitrary t > 0.
We also assume the integrability condition
∞
(H s ) :
d x (1 + |x|)s {|q(−x) − ql | + |q(x) − qr |} < +∞,
(2.3)
0
where s = 0, 1, 2 depending on the situation [and, again, the relevant condition will be assumed to
hold for all t ≥ 0]. Note that the condition (H s ) is equivalent to assuming (q(x) − ql/r ) ∈ L 1,s (R∓ ).
For later convenience, we denote the limits of Q(x, t) as x → +∞ and x → −∞ as Q r (t) and
Q l (t), respectively. We also introduce the “free” potential matrix Q f (x, t) as follows:
Q f (x, t) = Q r (t) θ (x) + Q l (t) θ (−x),
(2.4)
where θ (x) is the Heaviside function [as specified above, in the following the time dependence of
Q f and Q l/r will be omitted for brevity.]
It is convenient to introduce asymptotic scattering operators as x → ∓∞,
l/r (k) = −ikσ3 + Q l/r ,
(2.5)
(x, k) = −ikσ3 + Q f (x) = r (k)θ (x) + l (k)θ (−x) ,
(2.6)
˜
˜
and define the fundamental eigensolutions (x,
k) and (x,
k) as those 2 × 2 matrix solutions to
the first of (2.1) which satisfy the asymptotic conditions
˜
(x,
k) = e x r (k) [I2 + o(1)]
x → +∞ ,
(2.7a)
˜
(x,
k) = e x l (k) [I2 + o(1)]
x → −∞ .
(2.7b)
[Here and in the following I2 denotes the 2 × 2 identity matrix]. Note that e x l/r (k) is a bounded
group for all x ∈ R iff l/r (k) has only zero or purely imaginary eigenvalues and is diagonalizable.
This occurs iff k ∈ R ∪ (−i Al/r , i Al/r ), respectively. For k = ±i Al/r , the norm of the group e x l/r (k)
grows linearly in x as x → ∓∞ (see the Appendix for the explicit computation of the relevant norm
estimates for the groups e x l/r (k) ). Then the following result can be established similarly as to what
was done for the defocusing NLS equation in a preceding paper.18 For convenience we sketch the
proof in the Appendix.
Proposition 2.1. Let the potential satisfy (H 0 ). Then for k ∈ R ∪ (−i Ar , i Ar ) the fundamental
˜
eigensolution (x,
k) with asymptotic behavior (2.7a) can be obtained as the unique solution to the
integral equation
∞
˜
˜
dy e(x−y)r (k) [Q(y) − Q r ](y,
k) .
(2.8a)
(x,
k) = e x r (k) −
x
˜
Moreover, (x,
k) is continuous for x0 ≤ x for any finite x0 , and, as a function of k, for all
˜
k) is
k ∈ R ∪ (−i Ar , i Ar ). Similarly, for k ∈ R ∪ (−i Al , i Al ) the fundamental eigensolution (x,
given by the unique solution to the integral equation
x
˜
˜
dy e(x−y)l (k) [Q(y) − Q l ]
(y,
k) ,
(2.8b)
(x,
k) = e x l (k) +
−∞
continuous for x ≤ x0 for any finite x0 , and for all k ∈ R ∪ (−i Al , i Al ). In addition, under the hypothesis (H 1 ), (2.8a) has a unique, continuous solution for k ∈ [−i Ar , i Ar ], and (2.8b) has a unique,
continuous solution for k ∈ [−i Al , i Al ] (i.e., the result can be extended up to the corresponding
branch points).
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Assuming (H 1 ), one can replace the integral equations (2.8a)–(2.8b) by different ones. Let us
introduce the fundamental matrix G(x, y; k) as follows:
G(x, y; k) = θ (x)θ (y) e(x−y)r (k) + θ (−x)θ (−y) e(x−y)l (k)
+ θ (x)θ (−y) e
xr (k) −yl (k)
e
+ θ (−x)θ (y) e
(2.9)
xl (k) −yr (k)
e
.
G(x, y; k) is a continuous matrix function of (x, y, k) ∈ R2 × C which satisfies the initial value
problems
∂G(x, y; k)
= (x, k)G(x, y, k) ,
∂x
∂G(x, y; k)
= −G(x, y, k)(y, k) ,
∂y
G(y, y; k) = I2 ,
G(x, x; k) = I2 ,
(2.10a)
(2.10b)
where (x, k) is given by (2.6). For further details on the fundamental matrix we refer to Appendix A
of Ref. 18, where their properties were investigated for the defocusing NLS equation. Then using
(2.10a)–(2.10b), one can easily check that the fundamental eigenfunctions also satisfy the integral
equations
∞
˜
˜
dy G(x, y; k)[Q(y) − Q f (y)](y,
k),
(2.11a)
(x,
k) = G(x, 0; k) −
˜
(x,
k) = G(x, 0; k) +
x
x
−∞
˜
dy G(x, y; k)[Q(y) − Q f (y)]
(y,
k),
(2.11b)
with Q f (x) as in (2.4), and G(x, 0; k) = θ (x) e xr (k) + θ (−x) e xl (k) according to (2.9). Note that
(2.11a) coincides with (2.8a) for x ≥ 0, and (2.11b) coincides with (2.8b) for x ≤ 0. On the other
hand, using (2.9) we get
∞
xl (k)
˜
˜
I2 −
dy G(0, y; k)[Q(y) − Q f (y)](y, k)
x ≤ 0,
(2.12a)
(x, k) = e
˜
(x,
k) = e xr (k) I2 +
x
x
−∞
˜
dy G(0, y; k)[Q(y) − Q f (y)]
(y,
k)
x ≥ 0,
(2.12b)
where, unlike (2.8a)–(2.8b), the integrals in the right-hand sides converge absolutely as x → ∓∞.
Note, however, that since the fundamental matrix G(x, y; k) depends on both groups e xl/r (k) , the
˜
integral equations (2.11) can only be used to define both fundamental eigenfunctions (x,
k) and
˜
(x, k) for k ∈ R ∪ [−i Al , i Al ].
Finally, note that any matrix solution of the scattering problem (2.1) satisfies ∂∂x det M =
trace(−ikσ3 + Q) det M = 0. Consequently, the fundamental eigensolutions have determinants
which are independent of x, and from their asymptotic behavior (2.7a)–(2.7b) it follows that
˜
˜
det (x,
k) = det (x,
k) = 1 .
(2.13)
A. Jost solutions
In this section, we define the Jost solutions and derive their continuity and analyticity properties.
2
Since the asymptotic scattering operators l/r (k) are traceless, and such that l/r
(k) = −(k 2 +
2
Al/r )I2 , it is natural to consider the conformal mappings
λl = k 2 + Al2 ,
λr = k 2 + Ar2 ,
(2.14)
with branch cuts along the imaginary segments l/r = [−i Al/r , i Al/r ]. Introducing appropriate
local polar coordinates, we define
√
√
(2.15)
λr = r1r2 ei(θ1 +θ2 )/2 , λl = r3r4 ei(θ3 +θ4 )/2 ,
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√
r1 r2 ei with r1 = |k − i Ar |, r2 =
√
|k + i Ar | and = (θ1 + θ2 )/2 and angles −π/2 < θ1 , θ2 ≤ 3π/2. Similarly, λl = r3 r4 ei with r3 = |k − i Al |, r4 =
|k + i Al |, and = (θ3 + θ4 )/2 and angles −π/2 < θ3 , θ4 ≤ 3π/2.
FIG. 1. The branch cuts for λr =
k 2 + Ar2 and λl =
k 2 + Al2 : we define λr =
with r j ≥ 0, and −π/2 ≤ θ j < 3π/2 for j = 1, . . . , 4 as indicated in Fig. 1. We will consider a
single sheet of the complex plane for k, and denote by Kl/r the plane cut along the segments l/r
±
on the imaginary axis. C ± will denote the open upper/lower complex half planes, and Kr/l
the open
upper/lower complex half-planes cut along r/l , respectively. Then the following results can easily
be established.
λr provides one-to-one correspondences between the following sets:
•
•
•
•
k
k
k
k
∈ Kr+ ≡ C + \ (0, i Ar ] and λr ∈ C +
∈ ∂Kr+ ≡ R ∪ {is − 0+ : 0 < s < Ar } ∪ {i Ar } ∪ {is + 0+ : 0 < s < Ar } and λr ∈ R
∈ Kr− ≡ C − \ [−i Ar , 0) and λr ∈ C −
∈ ∂Kr− ≡ R ∪ {is − 0+ : −Ar < s < 0} ∪ {−i Ar } ∪ {is + 0+ : −Ar < s < 0} and λr ∈ R.
Similarly, λl provides one-to-one correspondences between the following sets:
•
•
•
•
k
k
k
k
∈ Kl+ ≡ C + \ (0, i Al ] and λl ∈ C +
∈ ∂Kl+ ≡ R ∪ {is − 0+ : 0 < s < Al } ∪ {i Al } ∪ {is + 0+ : 0 < s < Al } and λl ∈ R
∈ Kl− ≡ C − \ [−i Al , 0) and λl ∈ C −
∈ ∂Kl− ≡ R ∪ {is − 0+ : −Al < s < 0} ∪ {−i Al } ∪ {is + 0+ : −Al < s < 0} and λl ∈ R.
Note that with this choice for the branch cuts, one has λr ∼ λl ∼ k as k → ∞ in the entire
plane (cf. (A11)). In the following, λl± (k) will denote the boundary values taken by λl (k) for k ∈ l
from the right/left edge of the cut, with
±
λl (k) = ± Al2 − |k|2 ,
k = is ± 0+ , |s| ≤ Al
(2.16a)
on the right/left edge, and λr± (k) will denote the boundary values taken by λr (k) for k ∈ r from the
right/left edge of the cut, with
k = is ± 0+ , |s| ≤ Ar
(2.16b)
λr± (k) = ± Ar2 − |k|2 ,
on the right/left edge (cf. Fig. 1).
Clearly, ±iλl/r are the eigenvalues of l/r (k), and the eigenvector matrices Wl/r (k), such that
l/r (k)Wl/r (k) = −iλl/r Wl/r (k)σ3 ,
(2.17)
can be conveniently chosen as follows:
Wl/r (k) = I2 −
i
λl/r + k
σ3 Q l/r .
(2.18)
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We can then define the Jost solutions in terms of the fundamental eigensolutions (2.7a)–(2.7b) as
follows:
˜
(2.19a)
(x, k) = φ(x, k) φ̄(x, k) := (x,
k)Wl (k) ,
˜
(x, k) = ψ̄(x, k) ψ(x, k) := (x,
k)Wr (k) ,
(2.19b)
or, equivalently, as those solutions to the scattering problem in (2.1) with the following asymptotic
behavior:
(x, k) ∼ Wl (k) e−iλl xσ3
x → −∞ ,
(2.20a)
(x, k) ∼ Wr (k) e−iλr xσ3
x → +∞ .
(2.20b)
The Jost solutions ψ̄(x, k), ψ(x, k) are then defined for λr ∈ R, corresponding to k ∈ ∂Kr+ ∪
and when k = is ∈ [−i Ar , i Ar ] we will label with a superscript ± the values on the right/left
edge of the cut in both half-planes, i.e.,
˜
is) Wr (is ± 0+ )
|s| ≤ Ar ,
(2.21a)
± (x, is) ≡ ψ̄ ± (x, is) ψ ± (x, is) := (x,
∂Kr− ,
˜
since (x,
k) is single-valued across the cut, and Wr (k) has right/left limits defined via (2.16b).
Analogously, the Jost solutions φ(x, k), φ̄(x, k) are defined for λl ∈ R, corresponding to k ∈
∂Kl+ ∪ ∂Kl− , and when k = is ∈ [−i Al , i Al ] a superscript ± will be used to denote the values on
the right/left edge of the cut in both half-planes, i.e.,
˜
is) Wl (is ± 0+ )
|s| ≤ Al ,
(2.21b)
± (x, is) ≡ φ ± (x, is) φ̄ ± (x, is) := (x,
˜
since (x,
k) is single-valued across the cut, and Wl (k) has right/left limits defined via (2.16a).
The following theorems establish the analyticity properties of the Jost solutions as functions of
the scattering parameter k. The standard over-bar notation is used over the symbol of a set to denote
its closure.
Proposition 2.2. Suppose (H 1 ) holds. Then, for every x ∈ R, the Jost solution ψ(x, k) [resp.
ψ̄(x, k)] extends to a function that is continuous for k ∈ Kr+ ∪ ∂Kr− [resp. k ∈ Kr− ∪ ∂Kr+ ] and
analytic for k ∈ Kr+ [resp. k ∈ Kr− ]. Similarly, the Jost solution φ(x, k) [resp. φ̄(x, k)] extends to
a function that is continuous for k ∈ Kl+ ∪ ∂Kl− [resp. k ∈ Kl− ∪ ∂Kl+ ] and analytic for k ∈ Kl+
[resp. k ∈ Kl− ].
Proposition 2.3. Suppose (H 2 ) holds. Then, for every x ∈ R, the derivatives ∂k ψ(x, k),
∂k ψ̄(x, k), ∂k φ(x, k), and ∂k φ̄(x, k) extend to functions that are continuous for: k ∈ Kr+ ∪ ∂Kr− \
{−i Ar }, k ∈ Kr− ∪ ∂Kr+ \ {i Ar }, k ∈ Kl+ ∪ ∂Kl− \ {−i Al }, and k ∈ Kl− ∪ ∂Kl+ \ {i Al }, respectively; and analytic for: k ∈ Kr− , k ∈ Kr+ , k ∈ Kl− , and k ∈ Kl+ , respectively.
±
Note that Kr/l
are intended as analytic manifolds, and continuity of the Jost solutions across
the cuts is intended as the existence of right/left continuous limits only in the domains that have the
branch cut as part of their boundary as an analytic manifold. In the half-planes where locally there
is no analytic continuation off the branch cut, the functions ψ ± (x, k), ψ̄ ± (x, k) [resp. φ ± (x, k) and
φ̄ ± (x, k)] are as given in (2.21a) [resp. (2.21b)] with the two choices of λr± [resp. λl± ], and can be
obtained as the unique solutions of the corresponding Volterra integral equations.
B. Scattering coefficients
From the integral equations for the fundamental matrices (2.12a)–(2.12b), one can easily find
˜
(x,
k) = e xl (k) [Br (k) + o(1)]
x → −∞,
(2.22a)
˜
(x,
k) = e xr (k) [Bl (k) + o(1)]
x → +∞,
(2.22b)
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where the coupling matrices
J. Math. Phys. 55, 101505 (2014)
Br (k) = I2 −
−∞
Bl (k) = I2 +
∞
∞
−∞
˜
dy G(0, y; k)[Q(y) − Q f (y)](y,
k),
(2.23a)
˜
dy G(0, y; k)[Q(y) − Q f (y)]
(y,
k),
(2.23b)
are each other’s inverses. Under the assumptions of Proposition 2.1, in Eqs. (2.22a)–(2.22b)
and (2.23a)–(2.23b), one needs to take k ∈ R ∪ [−i Al , i Al ], where (2.11a)–(2.11b) are defined.
Furthermore, Eqs. (2.13), (2.22a)–(2.22b), and (A1) imply that det Br (k) = det Bl (k) = 1 for
k ∈ R ∪ [−i Al , i Al ].
Using (2.19a)–(2.19b) and (2.22a)–(2.22b) to obtain the asymptotic behavior of the Jost solutions
as x → ±∞, we can then express each set of Jost solutions as a linear combination of the other set,
i.e.,
φ(x, k) φ̄(x, k) = ψ̄(x, k) ψ(x, k) S(k),
(2.24a)
ψ̄(x, k)
ψ(x, k) = φ(x, k)
φ̄(x, k) S̄(k),
(2.24b)
where the scattering matrices S(k) and S̄(k) are obviously each other’s inverse, and they are given
by
S̄(k) = Wl−1 (k)Br (k)Wr (k).
S(k) = Wr−1 (k)Bl (k)Wl (k) ,
For later convenience, we write
a(k) b̄(k)
S(k) =
,
b(k) ā(k)
S̄(k) =
c̄(k) d(k)
,
d̄(k) c(k)
(2.25)
(2.26)
where the entries of the scattering matrices are usually referred to as scattering coefficients. It is clear
from (2.24a)–(2.24b) that S(k) [resp. S̄(k)] is in general defined where all four Jost solutions are,
i.e., for k ∈ ∂Kl− ∪ ∂Kl+ [resp. k ∈ ∂Kl− ∪ ∂Kl+ \ {±i Al }, the branch points being excluded because
of the second of (2.25)]. When k ∈ [−i Al , i Al ], the above scattering matrices and their entries are
defined by means of the corresponding values on the right/left edge of the cut, and labeled with
superscripts ± as clarified below. Since det (x, k) and det (x, k) are independent of x [and hence
these determinants can be computed from their limits as x → ∓∞], one can easily verify that
det S(k) =
λl (λr + k)
det Wl (k)
=
,
det Wr (k)
λr (λl + k)
det S̄(k) =
det Wr (k)
λr (λl + k)
=
,
det Wl (k)
λl (λr + k)
k ∈ R,
(2.27a)
and
det S ± (k) =
λl+ (k)(λr± (k) + k)
,
λr+ (k)(λl± (k) + k)
det S̄ ± (k) =
λr+ (k)(λl± (k) + k)
,
λl+ (k)(λr± (k) + k)
k ∈ (−i Al , i Al ) , (2.27b)
where we have taken into account (2.16a) and (2.16b).
def
If we now denote by Wr(v1 , v2 ) = det v1 v2 , the Wronskian of any two vector solutions v1 ,
v2 of the first of (2.1), then Eqs. (2.24a)–(2.24b) yield the following “Wronskian” representations
for the scattering coefficients:
a(k) =
λr + k
Wr(φ, ψ)
=
Wr(φ, ψ),
2λr
Wr(ψ̄, ψ)
ā(k) =
λr + k
Wr(ψ̄, φ̄)
=
Wr(ψ̄, φ̄),
2λr
Wr(ψ̄, ψ)
(2.28a)
b(k) =
λr + k
Wr(ψ̄, φ)
=
Wr(ψ̄, φ),
2λr
Wr(ψ̄, ψ)
b̄(k) =
λr + k
Wr(φ̄, ψ)
=
Wr(φ̄, ψ),
2λr
Wr(ψ̄, ψ)
(2.28b)
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and
λr (λl + k)
λl + k
a(k),
Wr(φ, ψ) =
2λl
λl (λr + k)
c̄(k) =
λl + k
λr (λl + k)
ā(k) ,
Wr(ψ̄, φ̄) =
2λl
λl (λr + k)
(2.28c)
λl + k
λr (λl + k)
b̄(k),
Wr(ψ, φ̄) = −
2λl
λl (λr + k)
d̄(k) =
λl + k
λr (λl + k)
Wr(φ, ψ̄) = −
b(k) ,
2λl
λl (λr + k)
(2.28d)
c(k) =
d(k) =
where the arguments (x, k) of the Jost solutions have been omitted for brevity, and the second set
of identities in each of (2.28c) and (2.28d) are obtained from S̄(k) = S −1 (k). Note that the above
Wronskian representations can be used to define the values of the scattering coefficients from the
right/left edge of the cuts l/r , consistently with (2.21a)–(2.21b) and (2.24a)–(2.24b). Explicitly,
one has
λ± (k) + k
Wr(φ ± (x, k), ψ ± (x, k))
k ∈ (−i Al , i Ar ) ,
a ± (k) = r ±
2λr (k)
ā ± (k) =
λr± (k) + k
Wr(ψ̄ ± (x, k), φ̄ ± (x, k))
2λr± (k)
k ∈ (−i Ar , i Al ) ,
b± (k) =
λr± (k) + k
Wr(ψ̄ ± (x, k), φ ± (x, k))
2λr± (k)
k ∈ (−i Al , i Ar ) ,
b̄± (k) =
λr± (k) + k
Wr(φ̄ ± (x, k), ψ ± (x, k))
2λr± (k)
k ∈ (−i Ar , i Al ) ,
and similarly for the scattering coefficients from the left.
Now Eqs. (2.28a)–(2.28d) allow one to extend some of the scattering coefficients under the
hypothesis (H 1 ). In fact, (2.28a)–(2.28d) and Proposition 2.2 imply:
• a(k) is continuous for k ∈ Kr+ ∪ ∂Kl− \ {i Ar } [with values across the cut denoted as a ± (k)],
and analytic in k ∈ Kr+ , while
a(k) ∼
i Ar
Wr(φ(x, i Ar ), ψ(x, i Ar )),
2λr
k → i Ar .
(2.29a)
• ā(k) is continuous in k ∈ Kr− ∪ ∂Kl+ \ {−i Ar } [with values across the cut denoted as ā ± (k)]
and analytic in k ∈ Kr− , while
ā(k) ∼
−i Ar
Wr(ψ̄(x, −i Ar ), φ̄(x, −i Ar )),
2λr
k → −i Ar .
(2.29b)
• b(k) is continuous for ∂Kr+ ∪ ∂Kl− \ {i Ar } [with values across the cut denoted as b± (k)], and
b̄(k) is continuous for k ∈ ∂Kr− ∪ ∂Kl+ \ {−i Ar } [with values across the cut denoted as b̄± (k)];
at the branch points where they are defined
b± (k) ∼
b̄± (k) ∼
i Ar
Wr(ψ̄(x, i Ar ), φ(x, i Ar )),
2λr
−i Ar
Wr(φ̄(x, −i Ar ), ψ(x, −i Ar )),
2λr
k → i Ar ,
k → −i Ar .
(2.29c)
(2.29d)
Similar results can be derived for the other four scattering coefficients, although the corresponding properties can also be obtained from those above using (2.28c) and (2.28d). Note
Eqs. (2.29a)–(2.29d) show that the scattering coefficients generically have singularities at the branch
points k = ±i Ar , i.e., for λr = 0. The behavior of the scattering coefficients at the branch points
will be further discussed in Sec. II D.
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For future convenience, we also define the reflection coefficients from the right as follows:
ρ(k) =
b(k)
a(k)
for k ∈ R,
ρ ± (k) =
b± (k)
a ± (k)
for k ∈ [−i Al , i Ar ) ,
(2.30a)
ρ̄(k) =
b̄(k)
ā(k)
for k ∈ R,
ρ̄ ± (k) =
b̄± (k)
ā ± (k)
for k ∈ (−i Ar , i Al ] ,
(2.30b)
and the reflection coefficients from the left as
r (k) =
d(k)
b̄(k)
≡−
c(k)
a(k)
r̄ (k) =
d̄(k)
b(k)
≡−
c̄(k)
ā(k)
r ± (k) =
for k ∈ R,
b̄± (k)
d ± (k)
≡ ±
±
c (k)
a (k)
r̄ ± (k) =
for k ∈ R,
for k ∈ [−i Al , i Al ] , (2.30c)
d̄ ± (k)
b± (k)
≡− ±
±
c̄ (k)
ā (k)
for k ∈ [−i Al , i Al ] .
(2.30d)
Note that in order to express r (k) and r̄ (k) in terms of the entries of S(k), we have used
that S(k) S̄(k) = I2 . The coefficients 1/a(k) [resp. 1/ā(k)] for k ∈ Kr+ [resp. for k ∈ Kr− ], and
1/a ± (k) [resp. 1/ā ± (k)] for k ∈ [−i Al , Ar ) [resp. k ∈ (−i Ar , i Al ] are usually referred to as (right)
transmission coefficients. Similarly definitions can obviously be introduced for the transmission
coefficients from the left, 1/c(k) and 1/c̄(k).
As in the defocusing case,18 from (2.25) and (2.23a)–(2.23b), using (2.17) and (2.19a)–(2.19b),
we can also obtain the following integral representation for the scattering matrix:
∞
dy eiλr yσ3 Wr−1 (k)[Q(y) − Q r ]
(y, k)
S(k) =
0
+
Wr−1 (k)Wl (k)
I2 +
0
dy e
−∞
iλl yσ3
Wl−1 (k)[Q(y)
− Q l ]
(y, k) ,
(2.31)
which could serve as an alternative to the Wronskian representations to establish the analytic
continuation in the appropriate half planes of the scattering coefficients a(k) and ā(k).
C. Symmetries of eigenfunctions and scattering data
∗
The scattering problem (2.1) admits two involutions: (k, λl/r ) → (k ∗ , λl/r
) and (k, λl/r ) →
(k, −λl/r ). Correspondingly, eigenfunctions and scattering data satisfy two sets of symmetry relations. In the asymmetric case treated here, where four branch points and two separate branch cuts
have to be considered, care must be taken in distinguishing between the case when both λr and λl
are discontinuous, which happens for k ∈ [−i Al , i Al ], and when only one is [here λr , because of
the choice Al ≤ Ar ], corresponding to k ∈ [−i Ar , −i Al ] ∪ [i Al , i Ar ].
First symmetry. On the single sheet for k we are considering, the involution k → k ∗ implies
∗
. One can easily check that if v(x, k) = (v1 (x, k) v2 (x, k))T [superscript T denotes matrix
λr/l → λr/l
transpose] is a solution of the ZS system (2.1), then v̂ ∗ (x, k) = iσ2 v(x, k ∗ ), σ2 being the second
Pauli matrix as given in (2.2), is a solution of the ZS system (2.1) as well. Taking into account the
boundary conditions (2.20a)–(2.20b), the symmetries for the Jost solutions are:
ψ̄ ∗ (x, k ∗ ) = iσ2 ψ(x, k) for k ∈ Kr+ ∪ R ,
ψ ∗ (x, k ∗ ) = −iσ2 ψ̄(x, k) for k ∈ Kr− ∪ R,
(2.32a)
(ψ̄ ± (x, k ∗ ))∗ = iσ2 ψ ± (x, k) for k ∈ [0, i Ar ],
(ψ ∗ ± (x, k ∗ ))∗ = −iσ2 ψ̄ ± (x, k) for k ∈ [−i Ar , 0],
(2.32b)
φ ∗ (x, k ∗ ) = iσ2 φ̄(x, k) for k ∈ Kl− ∪ R,
φ̄ ∗ (x, k ∗ ) = −iσ2 φ(x, k) for k ∈ Kl+ ∪ R, (2.32c)
(φ ± (x, k ∗ ))∗ = iσ2 φ̄ ± (x, k) for k ∈ [−i Al , 0] ,
(φ̄ ± (x, k ∗ ))∗ = −iσ2 φ ± (x, k) for k ∈ [0, i Al ] .
(2.32d)
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From (2.24a)–(2.24b) we then obtain S ∗ (k ∗ ) = σ2 S(k)σ2 wherever all entries in the scattering
matrix are simultaneously defined. In particular, under the assumption (H 1 ) for the potential, the
symmetry relations for the scattering coefficients can be written as
± ∗ ∗
ā ∗ (k ∗ ) =a(k) for k ∈ Kl+ ∪ R ,
ā (k ) = a ± (k) for k ∈ [−i Al , i Ar ) ,
(2.33a)
b̄∗ (k) = − b(k)
± ∗ ∗
b̄ (k ) = −b± (k) for k ∈ [−i Al , i Ar ) .
for k ∈ R ,
(2.33b)
It is worth noticing that the above symmetries relate the values of the scattering coefficients in the
upper/lower half plane of k, and from the same side of the cuts. Taking into account (2.30a)–(2.30d),
one can easily establish the symmetry relations satisfied by the reflection coefficients:
± ∗ ∗
ρ̄ (k ) = −ρ ± (k) for k ∈ [−i Al , i Ar ) ,
(2.33c)
ρ̄ ∗ (k) = −ρ(k) for k ∈ R ,
r̄ ∗ (k) = −r (k) for k ∈ R ,
± ∗ ∗
r̄ (k ) = −r ± (k) for k ∈ [−i Al , i Al ] .
(2.33d)
2
=
Second symmetry. When using a single sheet for the Riemann surface of the functions λl/r
2
2
k + Al/r , the involution (k, λl/r ) → (k, −λl/r ) can only be considered across the cuts. So this
second involution relates values of eigenfunctions and scattering coefficients for the same value of
k from either side of the cut. On the innermost cut, where both λl and λr are discontinuous, i.e., for
k ∈ l , one has
ψ̄ ∓ (x, k) =
λr± + k ±
ψ (x, k) ,
−iqr
φ̄ ∓ (x, k) =
λl± + k ±
φ (x, k)
−iql∗
for k ∈ [−i Al , i Al ] . (2.34)
For k ∈ r \ l , the symmetries for ψ, ψ̄ remain as above, while φ + (x, k) = φ − (x, k) for k ∈
[i Al , i Ar ] and φ̄ + (x, k) = φ̄ − (x, k) for k ∈ [−i Ar , −i Al ]. In addition, taking into account that
qr/l = Ar/l eiθr/l , the above symmetry relations also yield
ψ̄(x, ±i Ar ) = ∓e−iθr ψ(x, ±i Ar ) ,
φ̄(x, ±i Al ) = ∓eiθl φ(x, ±i Al ) .
(2.35)
Using the symmetries in the Wronskian representations for the scattering coefficients (2.26) one
obtains for k ∈ l ,
a ± (k) =
(λl± (k)
b± (k) =
ql∗ qr
qr (λl± (k) + k) ∓
ā ∓ (k) ≡
ā (k) ,
±
+ k)(λr (k) − k)
ql (λr± (k) − k)
ql∗ (λr± (k) + k) ∓
qr∗ (λl± (k) − k) ∓
b̄
b̄ (k) .
(k)
≡
qr (λl± (k) + k)
ql (λr± (k) − k)
(2.36a)
(2.36b)
On the other hand, for k ∈ r \ l the symmetry relations become
a ± (k) =
λr∓ (k) − k ∓
b (k)
iqr∗
for k ∈ [i Al , i Ar ] ,
(2.36c)
ā ∓ (k) =
λr∓ (k) + k ±
b̄ (k)
−iqr
for k ∈ [−i Ar , −i Al ] .
(2.36d)
Note that the above symmetries indeed hold also as k → i Ar for a(k), b(k) and as k → −i Ar
for ā(k), b̄(k), even when the scattering coefficients have singularities at these points (cf. (2.29a)–
(2.29d) and Sec. II D below for more details). Moreover, for k ∈ [i Al , i Ar ] using (2.36c) we have
ρ ± (k) =
a ∓ (k)
iqr∗
λr± (k) − k a ± (k)
⇒
a − (k)
λr+ (k) − k +
=
ρ (k) .
a + (k)
iqr∗
(2.37a)
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Note that the above relationships imply that ρ + (k) and ρ − (k) are related to each other by
ρ + (k)ρ − (k) = qr∗ /qr
for k ∈ [i Al , i Ar ] .
(2.37b)
Similarly, for k ∈ [−i Ar , −i Al ] (2.36d) yields
ρ̄ ± (k) =
−iqr ā ∓ (k)
λr∓ (k) + k ā ± (k)
ā + (k)
λr+ (k) + k −
ρ̄ (k)
=
ā − (k)
−iqr
⇒
(2.37c)
also implying
ρ̄ + (k)ρ̄ − (k) = qr /qr∗
for k ∈ [−i Ar , −i Al ] .
(2.37d)
For k ∈ r \ l , the symmetry relations for the scattering coefficients from the left are given
by
c± (k) =
−iqr
d̄ ∓ (k)
λr± (k) + k
for k ∈ [i Al , i Ar ] ,
(2.38a)
c̄± (k) =
λr∓ (k) + k ∓
d (k)
−iqr
for k ∈ [−i Ar , −i Al ] .
(2.38b)
D. Discrete eigenvalues, spectral singularities, and virtual levels
A discrete eigenvalue is a value of k ∈ Kr+ ∪ Kr− (corresponding to λr , λl ∈ C \ R) for which
there exists a nontrivial solution v to (2.1) with entries in L 2 (R). These eigenvalues occur for
k ∈ Kr+ iff the functions φ(x, k) and ψ(x, k) are linearly dependent (i.e., iff a(k) = 0), and for
k ∈ Kr− iff the functions ψ̄(x, k) and φ̄(x, k) are linearly dependent (i.e., iff ā(k) = 0). Equations
(2.20a)–(2.20b), together with the linear dependence requirement, imply that the corresponding
eigenfunctions are exponentially decaying as x → ±∞. The conjugation symmetry (2.33a) then
ensures that the discrete eigenvalues occur in complex conjugate pairs. The algebraic multiplicity
of each discrete eigenvalue coincides with the multiplicity of the corresponding zero of a(k) [for
k ∈ Kr+ ] or ā(k) [for k ∈ Kr− ]. In the following, we will assume discrete eigenvalues are simple, and
finite in number.
At each discrete eigenvalue kn ∈ K r+ , n = 1, . . . , N , the eigenfunctions φ(x, kn ) and ψ(x, kn )
are proportional, i.e., there exists a complex constant bn such that φ(x, kn ) = bn ψ(x, kn ). Then,
denoting by τn the residue of 1/a(k) at the (simple) pole λr = λr (kn ), we can write
lim (λr (k) − λr (kn ))
k→kn
φ(x, k)
= Cn ψ(x, kn ) ,
a(k)
Cn = bn τn ,
(2.39a)
and Cn is referred to as the norming constant associated with the discrete eigenvalue kn .
Similarly, at the (simple) discrete eigenvalues k1∗ , . . . , k ∗N in Kr− (necessarily also finite in
number, and the complex conjugates of the zeros of a(k) in Kr+ ) the eigenfunctions φ̄(x, kn∗ ) and
ψ̄(x, kn∗ ) are proportional to each other, i.e., there exist complex constants b̄n such that φ̄(x, kn∗ ) =
b̄n ψ̄(x, kn∗ ). Then, denoting by τ̄n the residue of 1/ā(k) at the pole λr = λr (kn∗ ) , we can write
lim∗ (λr − λr (kn∗ ))
k→kn
φ̄(x, k)
= C̄n ψ̄(x, kn∗ ) ,
ā(k)
C̄n = b̄n τ̄n ,
(2.39b)
and C̄n is referred to as the norming constant associated with the discrete eigenvalue kn∗ . Using the
symmetry relations (2.32a)–(2.32d) and (2.33a)–(2.33d), and the definitions (2.39a) and (2.39b), we
get
τ̄n = τn∗ ,
b̄n = −bn∗ ,
C̄n = −Cn∗ .
(2.40)
A spectral singularity is a value of k ∈ R for which a(k) = 0, or a value k ∈ [−Al , i Al ] for
which a ± (k) = 0 [similarly for ā(k) = 0, according to (2.33a) and (2.36a)–(2.36d)]. In this work,
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we assume no spectral singularities exist; also, we assume that a ± (k) = 0 for k ∈ r+ \ l+ (and,
correspondingly, ā ± (k) = 0 for k ∈ r− \ l− ). Unlike discrete eigenvalues, zeros of a ± (k) and ā ± (k)
is r \ l would not correspond to bound states for the eigenfunctions, since ψ(x, k) and ψ̄(x, k)
would not be exponentially decaying as x → +∞ for any such value of k. Note that according
to (2.37a) and (2.37c), requiring absence of spectral singularities implies that ρ ± (k) = 0 [resp.
ρ̄ ± (k) = 0] for all k ∈ r+ \ l+ [resp. k ∈ r− \ l− ]. Under these assumptions the transmission
coefficients 1/a ± (k) [resp. 1/ā ± (k)] and the reflection coefficients ρ ± (k) [resp. ρ̄ ± (k)] are continuous
in [−i Al , i Ar ] [ resp. [−i Ar , i Al ]. Similar statements hold for the transmission coefficients 1/c± (k),
1/c̄± (k), and reflection coefficients r ± (k), r̄ ± (k).
Establishing conditions on the asymptotic amplitudes and phases that guarantee absence of
spectral singularities, as well as of eigenvalues that lie in r \ l , is an interesting problem, but is
beyond the scope of this paper and will be the subject of future investigation. In any event, we mention
that spectral singularities can be incorporated in the inverse problem with slight modifications of the
approach presented in Sec. III.
Let us now investigate the behavior of the scattering coefficients at k = ±i Ar . While all eigenfunctions are continuous at these branch points (cf. Proposition 2.2), from the definition (2.28a)–
(2.28b), we expect the scattering coefficients to have singularities at the branch points of λr . We
say that the generic case holds if φ(x, i Ar ) and ψ(x, i Ar ) are linearly independent, or equivalently
[see (2.32a)–(2.32d)] iff ψ̄(x, −i Ar ) and φ̄(x, −i Ar ) are linearly independent. The exceptional case
holds if the above pairs of functions are linearly dependent, in which case k = ±i Ar are called
virtual levels (cf. Ref. 19, where this terminology is introduced for the defocusing NLS with equalamplitude BCs). In the generic case, the transmission coefficient 1/a(k) is continuous at k = i Ar
[equivalently, 1/ā(k) is continuous at k = −i Ar ]. In the exceptional case and under the hypothesis
(H 2 ) [cf. Theorem 2.3], we write (2.28a) in the form
λr + k Wr(φ, ψ) − Wr(φ, ψ)|k=i Ar
i Ar ∂
a(k) =
→
W (φ, ψ)
,
(2.41)
2
λr
2 ∂λr
k=i Ar
as k → i Ar . Thus a(k) is nonzero [hence, 1/a(k) is continuous] as k → i Ar iff the Wronskian
Wr(φ, ψ) has a simple zero at k = i Ar .
Note that the reflection coefficients ρ ± (k) and ρ̄ ± (k) are bounded as k → i Ar and k → −i Ar
respectively, both in the generic and in the exceptional case. In fact, using (2.35), from (2.29a) and
(2.29c) it follows that ρ ± (k) = b± (k)/a ± (k), is finite at k = i Ar ; in particular, in the generic case,
one has
ρ ± (k)|k=i Ar = e−iθr .
(2.42)
In the exceptional case, as long as the Wronskian Wr(φ, ψ) has a simple zero at k = i Ar , a(k)
is finite, and so ρ ± (k) are bounded at k = i Ar . Similarly, one can show that ρ̄ ± (k) is bounded
at k = −i Ar , and in the generic case ρ̄ ± (k)|k=−i Ar = eiθr . In particular, we note that one has
|ρ ± (i Ar )| = |ρ̄ ± (−i Ar )| = 1.
Obviously, similar considerations hold for the transmission coefficients from the left, 1/c(k),
1/c̄(k), and corresponding reflection coefficients r (k), r̄ (k). More specifically, from (2.28c) it follows
that c(k) and c̄(k) have the same zeros as a(k) and ā(k), respectively, and therefore the set of discrete
eigenvalues and (possibly) spectral singularities is completely determined by the latter.
The scattering coefficients from the left generically have singularities at the branch points
k = ±i Al (cf. (2.28c) and (2.28d)), unless the eigenfunctions become linearly dependent at these
points, which again will be referred to as the exceptional case. In both the generic and the exceptional
case (with a simple zero for the Wronskian of the eigenfunctions), the transmission and reflection
coefficients from the left are bounded at the branch points ±i Al .
E. Large k behavior of eigenfunctions and scattering data
We will derive the large k asymptotic expansion for the eigenfunctions from the Volterra integral
equations (A9a)–(A9d). Let us introduce for convenience the modified eigenfunction N (x, k) =
ψ(x, k) e−iλr x , and denote with a subscript j = 1, 2 its jth component. The integral equation (A9b)
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can be written explicitly in terms of the components of N (x, k) as follows:
iqr
λr + k ∞
N1 (x, k) = −
dy (q(y) − qr ) e2iλr (y−x) N2 (y, k)
−
λr + k
2λr
x
λr − k ∞
−
dy (q(y) − qr ) N2 (y, k)
2λr
x
∗
2iλr (y−x)
iqr ∞
iqr ∞
∗
+
dy q (y) − qr e
N1 (y, k) −
dy q ∗ (y) − qr∗ N1 (y, k) ,
2λr x
2λr x
∞
λr + k
N2 (x, k) = 1 +
dy q ∗ (y) − qr∗ N1 (y, k)
2λr
x
∗
λr − k ∞
+
dy q (y) − qr∗ e2iλr (y−x) N1 (y, k)
2λr
x
∗ ∞
iq
iq ∗ ∞
− r
dy (q(y) − qr ) N2 (y, k) + r
dy (q(y) − qr ) e2iλr (y−x) N2 (y, k) .
2λr x
2λr x
As a result of Proposition 2.2, in the above integral equations N1 (x, k) and N2 (x, k) − 1 are uniformly
bounded for (x, k) ∈ [x0 , +∞) × [Kr+ ∪ ∂Kr+ ∪ ∂Kr− ]. Also, the iteration of the integral equations
converges uniformly for (x, k) belonging to the same set. Assuming the potential is such that its
distributional derivative ∂x q ∈ L 1 (R), we can write
∞
∞
i
q(x) − qr +
dy (q(y) − qr )e2iλr (y−x) =
dy ∂ y q(y)e2iλr (y−x)
2λr
x
x
i q(x) − qr
+ o(1/k) .
(2.43)
2
k
Iterating once in the integral equation for N1 (x, k) [resp. N2 (x, k)] for the other unknown N2 (x, k)
[resp. N1 (x, k)], and using (2.43), yields for the inhomogeneous terms the following expansions in
powers of 1/k:
=
i[q(x) − qr ]
iq(x)
iqr
−
+ o(1/k) = −
+ o(1/k) ,
2k
2k
2k
∞
i
N2inh (x, k) = 1 −
dy (q ∗ (y) − qr∗ )q(y) + o(1/k) .
2k x
N1inh (x, k) = −
Substituting these expressions into the integral equations and computing their first iterates, it is clear
that only the last but one term in the right-hand side of the second integral equation leads to an
additional contribution of order 1/k, namely,
iq ∗ ∞
− r
dy (q(y) − qr ).
2k x
The iterations do not lead to any other terms of O(1) or O(1/k) [in particular, the last term in the
second integral equation is of higher order, because of Riemann-Lebesgue Lemma]. As a result, we
obtain
iq(x)
+ o(1/k) ,
(2.44a)
N1 (x, k) = −
2k
∞
iqr∗ ∞
i
dy (q(y) − qr ) − q(y)
dy (q ∗ (y) − qr∗ ) + o(1/k)
2k x
2k
x
∞
i
≡1−
dy (|q(y)|2 − Ar2 ) + o(1/k) .
(2.44b)
2k x
N2 (x, k) = 1 −
Note that in the above proof we have not assumed a priori the existence of an asymptotic
expansion for N (x, k) at large k. Instead, under the hypotheses (H 1 ) and ∂x q ∈ L 1 (R), the expansion
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follows automatically. Higher order asymptotic expansions [up to O(1/k n )] require assuming (H 1 )
j
and ∂x q ∈ L 1 (R) for j = 1, 2, . . . , n. Note also that ∂x q ∈ L 1 (R) implies that q is absolutely
continuous as a function of x, which is consistent with (2.44a). Moreover, the absolute continuity of
the potential and the assumption (H 0 ) ensure that the integral in (2.44b) is convergent as well. In a
similar way, one can determine the large k asymptotic behavior for the other eigenfunctions.
Summarizing, the Volterra integral equations (A9a)–(A9d) yield the following asymptotic behaviors for the eigenfunctions as |k| → ∞ in the appropriate half planes:
i Q(x)σ3
iλr σ3 x
(x, k)e
= I2 +
[1 + o(1)] ,
(2.45a)
2k
i Q(x)σ3
[1 + o(1)] .
(2.45b)
(x, k)eiλl σ3 x = I2 +
2k
For later convenience, we also observe that
i∂x Q(x)σ3
[1 + o(1)] .
(2.46)
∂x (x, k)eiλr σ3 x =
2k
From the Wronskian representations (2.28a)–(2.28b) for the scattering coefficients, and taking again
into account that λr ∼ λl ∼ k as k → ∞, we then obtain the asymptotic behavior of the scattering
coefficients
λr + k Wr φ(x, k) ψ(x, k) ∼ 1 as |k| → ∞, k ∈ Kr+ ∪ R,
(2.47a)
a(k) =
2λr
ā(k) = −
λr + k Wr φ̄(x, k)
2λr
ψ̄(x, k) ∼ 1 as
|k| → ∞, k ∈ Kr− ∪ R,
(2.47b)
while
b(k) = O(1/k 2 ) ,
b̄(k) = O(1/k 2 )
as
|k| → ∞, k ∈ R .
Taking into account (2.30a)–(2.30d), the above also imply that
ρ(k) = O(1/k 2 ) ,
ρ̄(k) = O(1/k 2 )
as
|k| → ∞, k ∈ R ,
(2.47c)
r (k) = O(1/k 2 ) ,
r̄ (k) = O(1/k 2 ) as
|k| → ∞, k ∈ R .
(2.47d)
F. Trace formula
In order to derive a representation for the scattering coefficient a(k) in terms of discrete eigenvalues and reflection coefficients, which is usually referred to as trace formula, the starting point is
the quasi-unitarity of the scattering matrix S(k). Taking into account the symmetries (2.33a)–(2.33d),
Eq. (2.27a)–(2.27b) for det S(k) becomes
|a(k)|2 + |b(k)|2 =
a ± (k)(a ± (k ∗ ))∗ + b± (k)(b± (k ∗ ))∗ =
λl (λr + k)
,
λr (λl + k)
k ∈ R,
λl+ (λr± + k)
,
λr+ (λl± + k)
k ∈ l .
In turn, the above equations can be written in terms of the reflection coefficients as follows:
|a(k)|2 =
a ± (k)(a ± (k ∗ ))∗ =
−1
λl (λr + k) 1 + |ρ(k)|2
,
λr (λl + k)
k ∈ R,
−1
λl+ (λr± + k) 1 + ρ ± (k)(ρ ± (k ∗ ))∗
,
+ ±
λr (λl + k)
(2.48a)
k ∈ l .
(2.48b)
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FIG. 2. Oriented contours C+ (left) and C− (right).
Since a(k) [resp. ā(k)] is analytic in Kr+ [resp. Kr− ] and continuous in Kr+ [resp. Kr− ], approaches
1 as k → ∞, and has (simple) zeros at k = kn , n = 1, · · · , N , [resp. k = kn∗ , n = 1, · · · , N ], we
introduce
α(k) = a(k)
N
k − k∗
n
n=1
k − kn
,
ᾱ(k) = ā(k)
N
k − kn
.
k − kn∗
n=1
(2.49)
Because of the analyticity properties of α(k) and ᾱ(k), Cauchy’s integral formula for k ∈ Kr+ yields
log α(ζ )
1
dζ
log α(k) =
2πi C+ ζ − k
0
0
i Al
1
log α(ζ )
log α − (ζ )
log α + (ζ )
log α − (ζ )
=
dζ −
dζ +
dζ +
dζ
2πi R ζ − k
ζ −k
ζ −k
ζ −k
−i Al
−i Al
0
i Ar
i Ar
i Al
log α − (ζ )
log α + (ζ )
log α + (ζ )
+
dζ −
dζ −
dζ ,
(2.50a)
ζ −k
ζ −k
ζ −k
i Al
i Al
0
0=
=
1
2πi
1
2πi
+
log ᾱ(ζ )
dζ
ζ −k
C−
R
0
−i Al
log ᾱ(ζ )
dζ −
ζ −k
0
−i Al
+
log ᾱ (ζ )
dζ +
ζ −k
i Al
0
log ᾱ − (ζ )
dζ −
ζ −k
−
log ᾱ (ζ )
dζ −
ζ −k
−i Al
−i Ar
i Al
0
log ᾱ − (ζ )
dζ +
ζ −k
log ᾱ + (ζ )
dζ ,
ζ −k
−i Al
−i Ar
log ᾱ + (ζ )
dζ
ζ −k
(2.50b)
where C± are the oriented contours illustrated in Fig. 2, and the superscripts ± in α(k), ᾱ(k) are
chosen depending on whether the integration is performed on the right or the left edge of each cut.
Adding the two equations and using the definitions of α(k) and ᾱ(k), we get
i Al log a − (ζ )ā − (ζ )
N
1
log[a(ζ )ā(ζ )]
k − kn
a + (ζ )ā + (ζ )
+
log a(k) =
log
dζ +
dζ
∗
k
−
k
2πi
ζ
−
k
ζ
−k
R
0
n
n=1
+
+
+
0
)ā (ζ )
log aa − (ζ
(ζ )ā − (ζ )
−i Al
ζ −k
dζ +
i Ar
i Al
−
)
log aa + (ζ
(ζ )
ζ −k
dζ +
−i Al
−i Ar
+
)
log āā − (ζ
(ζ )
ζ −k
dζ
.
(2.51)
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Using (2.48a)–(2.48b) and (2.37a), (2.37c), we arrive at the so-called trace formula
a(k) =
N k − kn
n=1
k − kn∗
1
log {γ (ζ ) [1 + ρ(ζ )ρ ∗ (ζ ∗ )]}
exp −
dζ
2πi ζ −k
+
i Ar log |λr (ζ )|−ζ
ρ
(ζ
)
∗
iqr
1
+
dζ
2πi i Al
ζ −k
⎫
−i Al log |λr (ζ )|+ζ ρ̄ − (ζ )
⎬
−iqr
1
+
dζ ,
⎭
2πi −i Ar
ζ −k
(2.52)
where γ (k) = [λr (λl + k)]/[λl (λr + k)] is a short-hand notation for 1/ det S(k) (cf. (2.27a)–(2.27b)),
and = R ∪ [−i Al , 0]left ∪ [−i Al , 0]right ∪ [0, i Al ]left ∪ [0, i Al ]right , oriented as in Fig. 2. Note in
particular that the term in square bracket in the numerator in the first integral of (2.52) becomes
[1 + |ρ(ζ )|2 ] for ζ ∈ R, and [1 + ρ ± (ζ )(ρ ± (ζ ∗ ))∗ ] for ζ ∈ [−i Al , i Al ]. Equation (2.52) shows that
a(k) is completely determined for k ∈ Kr+ in terms of: (i) its zeros (discrete eigenvalues) kn ∈ Kr+ ;
(ii) the reflection coefficient ρ(k) for k ∈ R, and ρ ± (k) for k ∈ l ; (iii) ρ + (k) for k ∈ [i Al , i Ar ],
and ρ̄ − (k) for k ∈ [−i Ar , −i Al ].
III. INVERSE SCATTERING PROBLEM
In the IST machinery, the inverse scattering problem consists of first reconstructing the eigenfunctions in terms of scattering data, and then obtaining the potential (i.e., the NLS solution) in terms
of the eigenfunctions. For instance, in formulating the inverse problem from the right, the following
set of scattering data is required: (i) the reflection coefficient ρ(k) for k ∈ R, and its values ρ ± (k) for
k ∈ [−i Al , i Al ] on either side of the cut [this accounts for the continuous spectrum of the scattering
operator, and it plays the role of the direct Fourier transform of the initial datum in the solution of the
initial-value problem for a linear PDE via Fourier transform; note that the reflection coefficient ρ̄(k)
is related to ρ(k) by the symmetry (2.33c)]; (ii) discrete eigenvalues kn ∈ Kr+ , and associated norming constants Cn , n = 1, · · · , N , as in (2.39a) [note that discrete eigenvalues in Kr− and associated
norming constants are not independent data, as they can be obtained from the above by conjugation
symmetry]; (iii) additional scattering data ρ + (k) for k ∈ [i Al , i Ar ] and ρ̄ − (k) for k ∈ [−i Ar , i Al ]
[note ρ − (k) and ρ̄ + (k) are related to the latter by symmetries (2.37b) and (2.37d); also note that
according to the trace formula in Sec. II F, the values of the transmission coefficient 1/a(k) for all
k ∈ Kr+ , and 1/a ± (k) for k ∈ r+ can be obtained from the above scattering data]. In this section,
we will formulate the inverse problem, i.e., the problem of reconstructing the eigenfunctions from
the scattering data, both in terms of (left and right) Marchenko equations, and as a Riemann-Hilbert
problems. As explained below, simple algorithms then allow one to obtain the potential from either
the Marchenko kernels, or the solution of the Riemann-Hilbert problem.
A. Triangular representations for the eigenfunctions
In this section, we establish the following two triangular representations for the fundamental
eigenfunctions:
∞
˜
ds K (x, s)e(s−x)r (k) ,
(3.1a)
(x,
k)e−xr (k) = I2 +
x
˜
(x,
k)e−xl (k) = I2 +
x
−∞
ds J(x, s)e(s−x)l (k) ,
(3.1b)
where the kernels K (x, s) = K i j (x, s) i, j=1,2 and J(x, s) = J i j (x, s) i, j=1,2 are “triangular” kernels, i.e., such that K (x, y) ≡ 0 for x > y, and J(x, y) ≡ 0 for x < y. We note that (3.1a)–(3.1b)
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Demontis et al.
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yield the corresponding triangular representations for the Jost solutions (2.19a)–(2.19b),
∞
−iλr σ3 x
+
ds K (x, s)Wr (k) e−iλr σ3 s ,
(x, k) = Wr (k)e
(3.2a)
x
(x, k) = Wl (k)e
−iλl σ3 x
+
x
−∞
ds J(x, s)Wl (k) e−iλl σ3 s .
(3.2b)
From the explicit expression of the groups e xr/l (k) in (A1), and taking into account that k → −k
corresponds to λr/l → −λr/l (cf. (2.15)), one can obtain from (3.1a), for x ∈ R and k ∈ R ∪
[−i Ar , i Ar ],
∞
1
i
λr
˜
k)e−xr (k) − I2 I2 − σ3 Q r − σ3
K (x, y) =
dλr e−iλr [y−x] (x,
(3.3a)
4π −∞
k
k
i
λr
−xr (−k)
˜
.
− I2 I2 + σ3 Q r + σ3
+ (x, −k)e
k
k
In a similar way, from the second of (3.1b), one obtains
∞
1
i
λl
−iλl [y−x]
−xl (k)
˜
J(x, y) =
(x, k)e
dλl e
− I2 I2 − σ3 Q r − σ3
4π −∞
k
k
i
λl
˜
.
+ (x,
−k)e−xl (−k) − I2 I2 + σ3 Q r + σ3
k
k
(3.3b)
The above inversion formulas for the Marchenko kernels will be derived in the Appendix. We
can now prove the existence of the kernels K (x, y) and J(x, y) as the Fourier transforms given by
(3.3a)–(3.3b). To show this, we need to assume that the potential q(x) satisfies (H 2 ) and ∂x q ∈ L 1 (R)
(which implies that q(x) is continuous in x ∈ R, and q(x) → qr/l as x → ±∞). If (3.3a) defines,
as a function of x, K (x, y) as the Fourier transform of a matrix function with entries in L 2 (R, dλr ),
we can state the existence of the kernel K (x, y). First of all, we observe that assuming (H 1 ) and
∂x q ∈ L 1 (R), we can write
i
λr − k
1 (1)
˜
N (x) + o(1) Wr−1 (k)
(x,
k)e−xr (k) − I2 =
σ3 Q r −
I2 +
2λr
2λr
k
1 (1)
i
N (x) + o(1) + O(1/k 2 ),
σ3 Q r +
=
2λr
k
(1)
(1)
iλr xσ3
where N(x, k) = (x, k) e
, and N (x) = N̄ (x) N (1) (x) is the O(1/k) term of the large
k asymptotic expansion of N(x, k) (cf. (2.19b) and Sec. II E). Similarly,
i
λr − k
1 (1)
˜
N (x) + o(1) Wr−1 (−k)
(x,
−k)e−xr (−k) − I2 = −
σ3 Q r −
I2 −
2λr
2λr
k
1 (1)
i
N (x) + o(1) + O(1/k 2 ).
σ3 Q r −
=−
2λr
k
We can now show that K (x, y) belongs to L 2 (x, +∞) in y. In order to do so, we need to prove that
∞
i
λr
−xr (k)
˜
(x,
k)e
σ
σ
I
dλr −
I
−
Q
−
2
2
3 r
3
k
k
−∞
2
i
λr
˜
+ (x,
−k)e−xλr (−k) − I2 I2 + σ3 Q r + σ3 k
k
is finite for each x ∈ R. Let us split up the integral into two parts to avoid the potentially bad
points k = 0 [i.e., λr = ±Ar ] and k = ±i Ar [i.e., λr = 0]. Under the hypothesis (H 1 ), the integral
with respect to λr over any neighborhood λr ∈ [−δ, δ] with 0 < δ < Ar avoids the point k = 0,
and the corresponding integrand function is continuous in λr , which leads to a well-behaved (and
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hence finite) integral. The integral with respect to λr over the remainder of the real λr -line involves
integration over a domain which does not bypass λr = ±Ar , i.e., k = 0. However, if we replace this
integral by an integral with respect to k, use that dλr = (k/λr )dk, assuming (H 2 ), the existence of
the limit
˜
˜
∂ ˜
−k)e−xλr (−k)
(x,
k)e−xr (k) − (x,
=
lim
(x, k)e−xr (k)
k→0
2k
∂k
k=0
[see Proposition 2.3] makes the integral well-behaved around k = 0. Thus the only thing left to
consider is the behavior as k → ±∞ [i.e., as λr → ±∞]. Specifically, it remains to be proved that,
say,
−1 ∞ 2
˜
k)e−xr (k) − I2 dk (x,
+
−∞
1
is finite for each x ∈ R. This indeed follows from the large k expansion in Sec. II E, since under the
assumptions (H 1 ) and ∂x q ∈ L 1 (R), the worst one can have as k → ±∞ is a term of order O(1/k).
We have therefore proved the existence of K (x, y) as a measurable function in y (for each x ∈ R),
satisfying
∞
dy K (x, y)2 < +∞.
x
Consequently, as a function of y, K (x, y) is the Fourier transform of an L 2 matrix function and
hence an L 2 matrix function itself, uniformly in x ≥ x0 for each x0 ∈ R. A similar argument can
be used for J(x, y). Finally, the following result allows one to recover the potential in terms of the
kernels K (x, y) and J(x, y).
Proposition 3.1. Suppose (H 2 ) holds, and in addition ∂x q ∈ L 1 (R). Then one has
Q(x) − Q r = 2 σ3 K (x, x)σ3 ,
Q(x) − Q l = 2 σ3 J(x, x)σ3 .
(3.4)
In particular, (3.4) shows that the diagonal entries of both Marchenko kernels K (x, x) and J(x, x)
are zero, while the off-diagonal entries satisfy the following relation:
q ∗ (x) = qr∗ − 2K 21 (x, x) = ql∗ + 2 J 21 (x, x) .
(3.5)
q(x) = qr − 2K 12 (x, x) = ql + 2 J 12 (x, x) ,
Assuming in addition q ∈ C 1 (R) ensures that (3.4) and hence (3.5) are defined everywhere, and
not merely almost everywhere.
Note that here we have omitted the time dependence for brevity. If all the above assumptions
on the potential hold for all t ≥ 0, then inserting the time dependence in the Jost and fundamental
eigenfunctions (see Sec. IV for details) yields a parametric t-dependence for the Marchenko kernels,
and the reconstruction formulas (3.5) for the potential for all t ≥ 0 read
q(x, t) = qr (t) − 2K 12 (x, x; t) = ql (t) + 2 J 12 (x, x; t) .
(3.6)
It is worth pointing out that the existence of the Marchenko kernels can be related to the following
Goursat problem
⎛
⎞
⎞ ⎛
K 11 (x, y)
−qr∗ q(x) K 12 (x, y) ⎠
⎠=⎝ ∗
(∂x + ∂ y ) ⎝
,
(3.7a)
K 22 (x, y)
−q (x) qr
K 21 (x, y)
⎛
(∂x − ∂ y ) ⎝
K 12 (x, y)
K 21 (x, y)
⎞
⎠=
−qr q(x)
−q ∗ (x) qr∗
K 11 (x, y)
K 22 (x, y)
,
(3.7b)
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with boundary conditions
q(x) = qr − 2K 12 (x, x) = qr + 2K ∗21 (x, x),
lim K jl (x, s) = 0,
s→∞
j, l = 1, 2 ,
(3.7c)
(3.7d)
which can be derived (under additional assumptions on the potential) from the scattering problem.
Here, however, we have provided an explicit inversion formula for the kernels that will appear in the
inverse problem in terms of the eigenfunctions of the direct scattering problem, and used Plancherel’s
theorem to find an explicit expression for K (x, y).
B. Marchenko equations
In this section, we derive (left and right) Marchenko integral equations as a means to solve the
inverse problem, i.e., reconstructing the eigenfunctions, and from them the potential, in terms of the
scattering data.
1. Right Marchenko equations
Let us write (2.24a) explicitly as
φ(x, k)
= ψ̄(x, k) + ρ(k)ψ(x, k)
a(k)
φ ± (x, k)
= ψ̄ ± (x, k) + ρ ± (k)ψ ± (x, k)
a ± (k)
φ̄(x, k)
= ψ(x, k) + ρ̄(k)ψ̄(x, k)
ā(k)
φ̄ ± (x, k)
= ψ ± (x, k) + ρ̄ ± (k)ψ̄ ± (x, k)
ā ± (k)
k ∈ R,
k ∈ [−i Al , i Ar ] ,
k ∈ R,
k ∈ (−i Ar , i Al ] ,
(3.8a)
(3.8b)
(3.8c)
(3.8d)
where ρ(k), ρ ± (k) and ρ̄(k), ρ̄ ± (k) are given by (2.30a) and (2.30b), respectively. It is important to
point out that even though Eq. (2.24a) is only defined for k ∈ R ∪ [−i Al , i Al ], Eq. (3.8b) shows the
first column of (2.24a) can be extended to k ∈ [i Al , i Ar ], and (3.8d) extends the second column of
(2.24a) to k ∈ [−i Ar , −i Al ]. In the following we will assume, in accordance with the discussion in
Sec. II D, that: (i) there are no spectral singularities; (ii) all discrete eigenvalues are simple; (iii) at
k = i Ar the Wronskian W (φ, ψ) does not have multiple zeros [recall the symmetries imply the same
holds at k = −i Ar for W (φ̄, ψ̄)]; (iv) (H 1 ) holds in the generic case and (H 2 ) in the exceptional
case.
Multiply (3.8a) by eiλr y for y > x, and substituting the triangular representations (3.2a), we
obtain
iλr x
∞
e φ(x, k)
ds K (x, s)Wr,1 (k) eiλr (y−s)
(3.9)
− Wr,1 (k) eiλr (y−x) =
a(k)
x
∞
ds K (x, s)Wr,2 (k) eiλr (s+y) ,
+ ρ(k) eiλr (x+y) Wr,2 (k) +
x
where Wr, j (k) denotes the jth column of the eigenvector matrix Wr (k) in (2.18). We recall that
λr ∼ λl as |k| → ∞, so that the term in the left-hand side decays as |k| → ∞ in Kr+ ∪ R. For the
purpose of this section, it will be convenient to consider the eigenfunctions as functions of λr , i.e.,
k = k(λr ) ≡ λr2 − Ar2 .
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FIG. 3. The oriented contours l± (left) and r± (right).
Note that λr ∈ R is in one-to-one correspondence with either k ∈ r+ or k ∈ r− (cf. Fig. 3). In the
following we will assume k ∈ r+ for the eigenfunction ψ(x, k) [analytic for k ∈ Kr+ ], and k ∈ r−
for ψ̄(x, k) [analytic for k ∈ Kr− ]. We then formally integrate (3.9) with respect to λr , exchange the
order of integration, and evaluate
∞
1
1
δ(y − s)
iλr (y−s)
dλr
=
e
−iqr∗ /(λr + k)
0
2π −∞
to obtain
1
I = K (x, y)
0
where
I≡
1
2π
∞
ds K (x, s)F(s + y),
(3.10)
x
∞
−∞
1
F(z) =
2π
+ F(x + y) +
∞
−∞
dλr
eiλr x φ(x, k)
− Wr,1 (k) eiλr (y−x) ,
a(k)
dλr ρ(k)Wr,2 (k) eiλr z .
As explained above, in the above integrals k = k(λr ) with k ∈ r+ . The next task is to express I
in terms of the Marchenko kernel K (x, y). Recall that we assumed that the discrete eigenvalues
k1 , . . . , k N , corresponding to the zeros of a(k) in Kr+ , are simple. Then, if we consider the function
g(λr ) obtained by subtracting to the integrand in I its poles, taking into account (2.39a), we obtain
g(λr ) = eiλr (y−x)
N
eiλr x φ(x, k)
eiλr (kn )y Cn ψ(x, kn )
− Wr,1 (k) −
.
a(k)
λr − λr (kn )
n=1
Since g(λr ) is analytic for λr ∈ C + , Residue Theorem and Jordan’s Lemma yield
I=i
N
eiλr (kn )y Cn ψ(x, kn ) .
n=1
Taking into account the triangular representation (3.2a), we get
∞
I = Fd (x + y) +
ds K (x, s)Fd (s + y),
x
where
Fd (z) = i
N
eiλr (kn )z Cn Wr,2 (kn ) .
n=1
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Substituting the above expression for I into (3.10), we then arrive at the right Marchenko integral
equation
∞
0
1
ds K (x, s)r (s + y) =
,
(3.11)
K (x, y)
+ r (x + y) +
0
0
x
where
r (z) = F(z) − Fd (z) =
1
2π
∞
−∞
N
dλr ρ(k)Wr,2 (k) eiλr z − i
eiλr (kn )z Cn Wr,2 (kn ) .
(3.12)
n=1
Again, note in the integral in (3.12) λr ∈ R, and correspondingly k = k(λr ) ∈ r+ .
Next, let us multiply (3.8c) by e−iλr y for y > x and substitute (3.2a)–(3.2b). We then obtain
−iλr x
∞
e
φ̄(x, k)
− Wr,2 (k) eiλr (x−y) =
ds K (x, s)Wr,2 (k) eiλr (s−y)
ā(k)
x
∞
+ ρ̄(k)e−iλr y e−iλr x Wr,1 (k) +
ds K (x, s)Wr,1 (k) e−iλr s .
x
Formally integrating with respect to λr and proceeding as before, we obtain
∞
0
Ī = K (x, y)
+
ds K (x, s) F̄(s + y) ,
1
x
where
1
Ī ≡
2π
F̄(z) =
∞
−∞
1
2π
dλr
∞
−∞
e−iλr x φ̄(x, k)
− Wr,2 (k) e−iλr (y−x) ,
ā(k)
dλr ρ̄(k)Wr,1 (k) e−iλr z .
In the above integrals, we assume k = k(λr ) ∈ r− (see Fig. 3). Taking into account (2.39b), exactly
as before, we can express Ī in terms of the Marchenko kernel
∞
ds K (x, s) F̄d (s + y) ,
Ī = F̄d (x + y) +
x
with
F̄d (z) = −i
N
∗
e−iλr (kn )z C̄n Wr,1 (kn∗ ) ,
n=1
thus arriving at the Marchenko integral equation
∞
0
¯ r (s + y) = 0 ,
¯ r (x + y) +
K (x, y)
ds K (x, s)
+
1
0
x
(3.13)
where
¯ r (z) = F̄(z) − F̄d (z) = 1
2π
∞
−∞
dλr e−iλr z ρ̄(k)Wr,1 (k)+ i
N
∗
e−iλr (kn )z C̄n Wr,1 (kn∗ ) .
(3.14)
n=1
Note that in the integral in (3.14) λr ∈ R, and k = k(λr ) ∈ r− .
As a result of the symmetries of the scattering data, one has
F ∗ (z) = iσ2 F̄(z),
Fd∗ (z) = iσ2 F̄d (z),
¯ r (z).
r∗ (z) = iσ2 (3.15)
In conclusion, we can write the Marchenko equations (3.11) and (3.13) as a single 2 × 2 Marchenko
equation with a 2 × 2 Marchenko kernel as follows:
∞
ds K (x, s)r (s + y) = 02×2 ,
(3.16)
K (x, y) + r (x + y) +
x
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where
¯ r (z) .
r (z) = r (z) (3.17)
¯ r are given by (3.12) and (3.14), and satisfy ¯ r (z) = iσ2 r∗ (z). Note that the 2 × 2 kernel
r , r (z) anticommutes with the Pauli matrix σ3 , and satisfies the conjugation symmetry relation
r∗ (z) = σ2 r (z)σ2 ,
(3.18)
in compliance with K ∗ (x, s) = σ2 K (x, s)σ2 .
2. Left Marchenko equations
In order to derive the left Marchenko equations, let us write (2.24b) explicitly as
ψ̄(x, k)
= φ(x, k) + r̄ (k)φ̄(x, k)
c̄(k)
k ∈ R,
ψ̄ ± (x, k)
= φ ± (x, k) + r̄ ± (k)φ̄ ± (x, k)
c̄± (k)
ψ(x, k)
= φ̄(x, k) + r (k)φ(x, k)
c(k)
k ∈ (−i Al , i Al ) ,
k ∈ R,
ψ ± (x, k)
= φ̄ ± (x, k) + r (k)± φ ± (x, k)
c± (k)
k ∈ (−i Al , i Al ) ,
(3.19a)
(3.19b)
(3.19c)
(3.19d)
where r (k), r ± (k) and r̄ (k), r̄ ± (k) are given by (2.30c) and (2.30d), respectively. Under the same
assumptions as in Sec. III B 1 regarding the potential and the discrete spectrum, and considering in
this case the eigenfunctions as functions of λl , with
k = k(λl ) = λl2 − Al2 ,
we multiply (3.19c) by e−iλl y for y < x and substitute the triangular representations (3.2b). Formally
integrating with respect to λl and exchanging the order of integration, we have
−iλl x
∞
∞
x
1
e
ψ(x, k)
1
− Wl,2 (k) eiλl (x−y) =
dλl
ds J(x, s)
dλl Wl,2 (k) eiλl (s−y)
2π −∞
c(k)
2π −∞
−∞
(3.20)
∞
∞
x
1
1
dλl r (k)Wl,1 (k) e−iλl (x+y) +
ds J(x, s)
dλl r (k)Wl,1 (k) e−iλl (s+y) ,
+
2π −∞
2π −∞
−∞
where Wl, j (k) denotes the jth column of the matrix of asymptotic eigenvectors Wl (k) (cf. (2.18)).
As previously noted, λl ∈ R is in one-to-one correspondence with either k ∈ l+ , or k ∈ l− (cf.
Fig. 3). We will consider k ∈ l+ for the eigenfunction φ(x, k) [analytic for k ∈ Kl+ ], and k ∈ l−
for φ̄(x, k) [analytic for k ∈ Kl− ]. As before, we can reduce the identity (3.20) to
x
0
ds J(x, s)G(s + y) ,
(3.21)
Ĩ = J(x, y)
+ G(x + y) +
1
−∞
where
Ĩ ≡
1
2π
−∞
1
G(z) =
2π
∞
dλl
∞
−∞
e−iλl x ψ(x, k)
− Wl,2 (k) eiλl (x−y) ,
c(k)
dλl r (k)Wl,1 (k) e−iλl z .
In order to compute Ĩ so as to express it in terms of the Marchenko kernel J(x, y), one needs to
be able to close the contour at infinity in the upper half-plane of λl . Unlike what happens for the
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FIG. 4. The contour (R, ε).
Marchenko equations from the right, in this case closing the contour at infinity requires including the
contribution of the additional branch cut that in the k-plane corresponds to r \ l . To this end, let
us consider, for 0 < ε < R < +∞, the closed contour (R, ε) consisting
of the following pieces,
2
2
with the orientation specified in Fig. 4: (i) [−R, −ε]; (ii) the segment −ε + i0, −ε + i Ar − Al
2
i[π−θ]
2
: 0 ≤ θ ≤ π clockwise
along the imaginary λl axis; (iii) the semicircle i Ar − Al + εe
oriented; (iv) the segment ε + i0, ε + i Ar2 − Al2 along the imaginary λl axis; (v) [ε, R]; (vi)
{Reiθ : 0 ≤ θ ≤ π } counterclockwise oriented. R is assumed large enough, and ε is small enough
so that for all of the finitely many discrete eigenvalues kn ∈ Kl+ , n = 1, 2, . . . , N , λl (kn ) belong to
the interior region of the contour. Since ψ(x, k) and 1/c(k) have finite limits as k → i Ar , the integral
defining Ĩ with the integration confined to the semicircle around the branch point does not contribute
as ε → 0+ . Because of Jordan’s lemma, the integral defining Ĩ when confined to the large semicircle
(vi) does not contribute either as R → +∞ [note λl ∼ λr as k → ∞, ensuring the integrand vanishes
as k → ∞]. Thus there are two nontrivial contributions to the integral, Ĩ = Ĩ1 + Ĩ2 : the contribution
Ĩ1 pertaining to the residues of the function under the
integral sign at the poles k ∈ Kr+ ; and the
contribution Ĩ2 pertaining to the integral around λl ∈ 0, i Ar2 − Al2 in the upper-half λl -plane.
We shall evaluate the two contributions separately.
Since we assumed that the discrete eigenvalues kn in Kr+ are simple poles of 1/c(k), and
the reflection and transmission coefficients are continuous for k ∈ ∂Kr+ , taking into account that
ψ(x, kn ) = φ(x, kn )/bn , we obtain
Ĩ1 = i
N
e−iλl (kn )y C̃n φ(x, kn ) ,
C̃n = τ̃n /bn ,
n=1
where τ̃n is the residue of 1/c(k) at λl = λl (kn ), and C̃n is the associated norming constant. Note
that (2.28c) implies the residues τ̃n and τn , and hence the norming constants C̃n and Cn , are related
as follows:
τ̃n =
λr (kn ) + kn
τn ,
λl (kn ) + kn
C̃n Cn = τn2
λr (kn ) + kn
.
λl (kn ) + kn
Therefore, we have
Ĩ1 = G 1 (x + y) +
x
−∞
ds J(x, s)G 1 (s + y) ,
(3.22)
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where
G 1 (z) = i
N
e−iλl (kn )z C̃n Wl,1 (kn ) .
n=1
2
2
Let us now look into the second contribution Ĩ2 , which arises for λl ∈ 0, i Ar − Al , correspond
ing to k ∈ [i Al , i Ar ] and λr ∈ R. Introducing = Ar2 − Al2 , we have
i− i+ ψ(x, k) −iλl x
1
dλl
e
−
− Wl,2 (k) eiλl (x−y)
Ĩ2 = lim
→0 2π
c(k)
i0−
i0+
−
i
ψ (x, k) ψ + (x, k) −iλl y
1
−
e
=
dλl
.
(3.23)
2π 0
c− (k)
c+ (k)
In the above integral along the cut [0, i] on the positive imaginary axis, as usual, superscripts ±
denote the limiting values from the right/left edge of the cut, respectively, and we have used that λl ,
and hence Wl,1 , are continuous across it. Using (2.28c), we can write Ĩ2 as
i
|λr | − k ψ − (x, k) |λr | + k ψ + (x, k)
λl
1
−
e−iλl y ,
dλl
Ĩ2 =
2π 0
|λr |
a − (k)
|λr |
a + (k)
λl + k
and the symmetry relations (2.34) and (2.36c) allow to express (|λr | − k)ψ − /a − = −(|λr | +
k)ψ̄ + /b+ , so that
i
λl (|λr | + k) ψ̄ + (x, k) ψ + (x, k) −iλl y
1
+
e
dλl
.
(3.24)
Ĩ2 = −
2π 0
|λr |(λl + k)
b+ (k)
a + (k)
Using first the scattering equation (3.8b), and then again the symmetry relation (2.36c), we finally
have
i
φ + (x, k) −iλl y
λl (|λr | + k) φ + (x, k) −iλl y iqr i
λl
1
dλl
=
dλ
.
e
e
Ĩ2 = −
l
2π 0
|λr |(λl + k) a + (k) b+ (k)
2π 0
|λr |(λl + k) a − (k)a + (k)
(3.25)
We can now insert into the last expression the triangular representation (3.2a), and obtain
x
ds J(x, s)G 2 (s + y) ,
(3.26)
Ĩ2 = G 2 (x + y) +
−∞
with
G 2 (z) =
iqr
2π
i
dλl
0
If we now define
1
l (z) =G(z) − G 1 (z) − G 2 (z) ≡
2π
−i
N
n=1
e
−iλl (kn )z
1
λl
Wl,1 (k) e−iλl z .
+
|λr |(λl + k) a (k)a − (k)
∞
−∞
iqr
C̃n Wl,1 (kn ) −
2π
dλl r (k)Wl,1 (k) e−iλl z +
i
dλl
0
(3.27)
1
λl
Wl,1 (k) e−iλl z ,
|λr |(λl + k) a + (k)a − (k)
where in the first integral, corresponding to λl ∈ R, one has k = k(λl ) ∈ l+ , and use (3.22) and
(3.26) to compute Ĩ = Ĩ1 + Ĩ2 and introduce it into (3.21), we finally arrive at the left Marchenko
integral equation
x
0
0
J(x, y)
+ l (x + y) +
ds J(x, s)l (s + y) =
.
(3.28)
1
0
−∞
In a similar way, starting from (3.19c), one can derive the “adjoint” left Marchenko equation
x
1
0
¯
¯
J(x, y)
+ l (x + y) +
ds J(x, s)l (s + y) =
,
(3.29)
0
0
−∞
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where
¯ l (z) = iσ2 l∗ (z) .
The two Marchenko equations can be written in a compact matrix form as follows:
J(x, y) + l (x + y) +
x
−∞
ds J(x, s)l (s + y) = 02×2 ,
(3.30)
where
¯ l (z) l (z) ,
l (z) = l∗ (z) = σ2 l (z)σ2 .
(3.31)
It is worth making some remarks on the left and right Marchenko integral equations (3.16) and
(3.30) derived above. First of all, notice that the asymmetry between left/right Marchenko equations
is due to the asymmetry of the BCs, and explicitly to the choice Ar ≥ Al (with Ar ≤ Al the roles
of the two integral equations is reversed). In the Marchenko integral equations from the left, l (z)
in (3.27) has three separate contributions: one from the discrete spectrum, one from the reflection
coefficients from the left, r (k) and r̄ (k), integrated over values of k in the continuous spectrum,
i.e., k ∈ R ∪ l , and a third contribution (sometimes referred to as the dispersive shock wave
contribution, or DSW) which contains an integral over imaginary values of λl where the product of
transmission coefficients 1/(a + (k)a − (k)) appears. On the other hand, r (z) in the integral equations
from the right (cf. Eq. (3.12)) has only two contributions: one from the discrete spectrum, and one
from the reflection coefficients from the right, ρ(k) and ρ̄(k). However, it should be noted that in
the latter the reflection coefficients are integrated over all λr ∈ R, which means that the integral
includes, in addition to the continuous spectrum R ∪ l , also a contribution from r \ l . Moreover,
the integrand over r \ l can never be identically zero, as, according to (2.37a), in the absence
of spectral singularities ρ ± (k) = 0 for all k ∈ [i Al , i Ar ], and ρ̄ ± (k) = 0 for all k ∈ [−i Ar , −i Al ].
In turn, this implies that when r \ l = ∅ (i.e., whenever one deals with asymmetric boundary
conditions with Ar = Al ), no pure soliton solutions exist.
The Marchenko integral equations obtained here provide the necessary setup for the study of
the long-time behavior of the solutions by means of matched asymptotics, as was recently done for
KdV in Ref. 1.
C. Riemann-Hilbert problem
The purpose of this section is to provide an alternate formulation of the inverse problem, which
is posed as a Riemann-Hilbert (RH) problem for the eigenfunctions, with jumps expressed in terms
of the scattering data. Once the RH problem is solved, the large k expansion of the eigenfunctions
then provides the reconstruction of the potential. We consider the following matrix of eigenfunctions:
⎧ φ(x,k)
⎨ a(k) eiλl x ψ(x, k) e−iλr x ,
M(x, k) = ⎩ ψ̄(x, k) eiλr x φ̄(x,k) e−iλl x ,
ā(k)
k ∈ Kr+
k ∈ Kr−
,
(3.32)
such that M(x, k) → I2 as k → ∞, and formulate the inverse problem as a Riemann-Hilbert problem
for the sectionally meromorphic matrix M(x, k) across ∂Kr+ ∪ ∂Kr− . Explicitly, we determine the
five jump matrices illustrated in Fig. 5: V0 is the jump matrix across the real axis of the complex
k-plane; V1 across l+ = [0, i Al ]; V2 across l− = [−i Al , 0); V3 across r+ \ l+ = (i Al , i Ar ], and
V4 across r− \ l− = [−i Ar , −i Al ). All jump matrices depend on k along the appropriate contour
in the complex plane, as well as, parametrically, on (x, t) ∈ R × R+ [the x-dependence is explicit,
while the time dependence is “hidden” in that of the corresponding reflection coefficients, see
Sec. IV, and will be omitted for brevity].
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FIG. 5. The jump matrices V j , j = 0, 1 · · · , 4, of the RH problem across R ∪ l+ ∪ l− ∪ (r+ \ l+ ) ∪ (r− \ l− ).
The RH problem across the real axis can be written in matrix form as: M + (x, k) =
M (x, k)V0 (x, k), i.e.,
+
φ (x, k) iλl x
φ̄ − (x, k) −iλl x
+
−iλr x
−
iλr x
= ψ̄ (x, k) e
V0 (x, k) ,
e
e
ψ (x, k) e
k ∈ R,
ā − (k)
a + (k)
(3.33)
where in this case the superscripts ± denote limiting values from the upper/lower complex plane,
respectively. The jump matrix across the real axis can be easily computed from (2.24a), and it is
given by
[1 − ρ(k)ρ̄(k)] ei(λl −λr )x −ρ̄(k) e−2iλr x
.
(3.34)
V0 (x, k) =
ρ(k) e2iλl x
ei(λl −λr )x
−
We then write the RH problem across l+ as: M + (x, k) = M − (x, k)V1 (x, k), k ∈ C + , where now
±
denote limiting values from the right/left edge of the cut across l+ (l in the upper half plane).
Taking into account that across l both λl and λr change sign, and using the notation
λl+ = −λl− = λl ,
we have
+
φ (x, k) iλl+ x
e
a + (k)
+
ψ (x, k) e
−iλr+ x
λr+ = −λr− = λr ,
φ − (x, k) iλl− x
=
e
a − (k)
−
ψ (x, k) e
−iλr− x
V1 (x, k),
(3.35)
and the jump matrix V1 can easily be computed using (2.24a) and the symmetry relations (2.34),
⎞
⎛
ei(λl −λr )x
ρ + (k) e2iλl x
iqr ⎝
⎠.
∗
V1 (x, k) = −
(3.36)
qr
λr + k
− ρ + (k)ρ − (k) ei(λl −λr )x −ρ − (k) e−2iλr x
qr
l−
The RH problem across
will be written as M + (x, k) = M − (x, k)V2 (x, k), k ∈ C − , with super±
scripts denoting non-tangential limits from the right/left of the cut across l− , i.e., l in the lower
half plane. Explicitly, one has
φ̄ + (x, k) −iλl+ x
φ̄ − (x, k) −iλl− x
+
−
iλr− x
=
V2 (x, k).
e
e
(x,
k)
e
(3.37)
ψ̄ + (x, k) eiλr x
ψ̄
ā + (k)
ā − (k)
As before, the jump matrix can be determined using (2.24a) and (2.34), and it is given by
⎞
⎛
qr
−
2iλr x
+
−
i(λr −λl )x
∗
(k)
e
−
ρ̄
(k)
ρ̄
(k)
e
−
ρ̄
∗
iqr ⎝
qr
⎠.
V2 (x, k) = −
λr + k
i(λr −λl )x
e
ρ̄ + (k) e−2iλl x
(3.38)
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The RH problem across r+ \ l+ is written as M + (x, k) = M − (x, k)V3 (x, k), k ∈ C + ,
−
+
φ (x, k) iλl x
φ (x, k) iλl x
+
−iλr+ x
− −iλr− x
=
V3 (x, k),
ψ (x, k) e
ψ e
e
e
a + (k)
a − (k)
(3.39)
and taking into account that λr changes sign, while λl and φ are continuous, from (2.24a) and (2.34),
one obtains
⎞
⎛
e−i(λl +λr )x
ρ + (k)
iqr ⎝
⎠.
∗
V3 (x, k) = −
qr
λr + k
− ρ + (k)ρ − (k) ei(λl −λr )x −ρ − (k) e−2iλr x
qr
The symmetry (2.37b) finally yields
⎛
V3 (x, k) = −
iqr ⎜
⎝
λr + k
ρ + (k)
q∗
− qrr
0
⎞
e−i(λl +λr )x
⎟
⎠.
1
−2iλr x
e
ρ + (k)
(3.40)
Finally, we write the RH problem across r− \ l− , where λr changes sign but λl and φ̄ are continuous,
as M + (x, k) = M − (x, k)V4 (x, k), k ∈ C − ,
φ̄ + (x, k) −iλl x
φ̄ − (x, k) −iλl x
+
iλr+ x
−
iλr− x
= ψ̄ (x, k) e
V4 (x, k),
e
e
(3.41)
ψ̄ (x, k) e
ā + (k)
ā − (k)
where
⎛
V4 (x, k) = −
iqr∗
λr + k
⎜
⎝
−ρ̄ − (k) e2iλr x
e
i(λr +λl )x
0
qr
qr∗
⎞
⎟
1 ⎠.
ρ̄ − (k)
(3.42)
Note that the jump matrices satisfy the following upper/lower half plane symmetry:
V2 (x, k) = σ2 V1∗ (x, k ∗ )σ2 ,
V4 (x, k) = σ2 V3∗ (x, k ∗ )σ2 .
Solving the inverse problem as a RH problem (with poles, corresponding to the zeros of a(k) and
ā(k) in the upper/lower half planes) then amounts to computing the sectionally meromorphic matrix
M(x, k) with the given jumps, and normalized to the identity as k → ∞. Specifically, we can write
the problem as M + = M − + (V − I2 )M − , where V (x, k) = V j (x, k) for j = 0, · · · , 4 depending
on which piece of the contour is being considered, and superscripts ± denote non-tangential limits
from either side of the contour. Then, subtracting the behavior as k → ∞, and the residues of M ±
at the poles in Kr± from both sides we obtain
M + − I2 −
N
n=1
M − − I2 −
N
n=1
N
1
1
Reskn M + −
Reskn∗ M − =
∗
k − kn
k
−
k
n
n=1
(3.43)
N
1
1
Reskn M + −
Reskn∗ M − + (V − I2 )M − .
∗
k − kn
k
−
k
n
n=1
The left-hand side of the above equation is now analytic in Kr+ , and it is O(1/k) as k → ∞ there,
while the sum of all terms but the last one in the right-hand side is analytic in Kr− , and is O(1/k) as
k → ∞ there. We then introduce projectors P± over r± ≡ R ∪ r± :
f (ξ )
1
P± [ f ](z) =
dξ ,
2πi r± ξ − k
%
%
where r+ [resp. r− ] denotes the integral along the oriented contours in Fig. 3, and when k ∈ r± ∩ R
the limit is taken from the above/below. One can easily prove that if f ± are analytic in Kr± and are
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O(1/k) as k → ∞, the following holds: P± f ± = ± f ± and P+ f − = P− f + = 0. Then, applying
P± to both sides of (3.43), we find
N
N
Reskn M + Reskn∗ M −
M − (ξ )
1
[V (ξ ) − I2 ] dξ , k ∈ C ± \ r ,
M(k) = I2 +
+
+
∗
± ξ − k
k
−
k
k
−
k
2πi
n
n
r
n=1
n=1
(3.44)
where the x-dependence in eigenfunctions and jump matrices has been omitted for brevity. Taking
into account that the second column of Reskn M + is zero for all n, while the first column is proportional
to the second column of M + (x, kn ), and vice-versa the first column of Reskn∗ M − is zero for all n,
while the second column is proportional to the second column of M − (x, kn∗ ) according to (2.39a)–
(2.39b), the above integral/algebraic system can be closed by evaluating it at each k = kn and k = kn∗ .
The potential is then reconstructed by the large k expansion of the latter, since
i
Q(x)σ3 [1 + o(1)] .
M(x, k) = I2 +
2k
Note that unlike what happens in the same-amplitude case, the above system cannot be reduced to a
purely algebraic one: although the reflection coefficients can be chosen to be identically zero on the
continuous spectrum, i.e., for k ∈ R ∪ l , the integrals appearing in the right-hand side of (3.44)
always exhibit a non-zero contribution from the contours r± \ l± . In particular, this implies that no
pure soliton solutions exist, and solitons are always accompanied by a radiative contribution of some
sort. One could nonetheless solve the system iteratively, assuming the reflection coefficients are
small for k ∈ l± (and/or for k ∈ r± \ l± ), and thus obtaining NLS solutions comprising solitons
superimposed to small radiation. Moreover, the RH problem formulated here provides the key
setup for the investigation of the long-time asymptotic behavior by the Deift-Zhou steepest descent
method.13, 14, 17 The time dependence in the system is simply accounted for by the time dependence
of the scattering coefficients, as described in Sec. IV. When one is interested only in capturing
the leading order behavior of the solution for large t, the jumps across the contours illustrated in
Fig. 5 and determined above can be simplified by suitable factorizations and contour deformations,
and reduced to certain model problems for which “explicit” solutions (often expressed in terms of
Riemann theta functions) can be sought for. This study obviously goes beyond the scope of the
present paper, and will be the subject of future investigation.
The RH problem could also be formulated in terms of left scattering data, introducing the
sectionally meromorphic matrix of eigenfunctions
⎧
e−iλr x , k ∈ Kr+
⎨ φ(x, k) eiλl x ψ(x,k)
c(k)
M̃(x, k) = ⎩ ψ̄(x,k) eiλr x φ̄(x, k) e−iλl x , k ∈ K− .
r
c̄(k)
The RH problem across the real axis can be written in matrix form as: M̃ + (x, k) =
M̃ (x, k)Ṽ0 (x, k), i.e., for k ∈ R,
−
ψ̄ (x, k) iλr x
ψ + (x, k) −iλr x
−
−iλl x
=
Ṽ0 (x, k) ,
e
e
φ + (x, k) eiλl x
(x,
k)
e
(3.45)
φ̄
c̄− (k)
c+ (k)
−
where in this case the superscripts ± denote limiting values from the upper/lower complex plane,
respectively. The jump matrix across the real axis can be easily computed from (2.24b), and it is
given by
i(λl −λr )x
e
r (k) e−2iλr x
Ṽ0 (x, k) =
.
(3.46a)
−r̄ (k) e2iλl x [1 − r (k)r̄(k)] ei(λl −λr )x
In a similar way, one can obtain the jump matrices Ṽ1 across l+ and Ṽ2 across l− , respectively,
⎞
⎛
ql
−
2iλl x
+
−
i(λl −λr )x
∗
(k)
e
−
r
(k)r
(k)
e
−r
∗
iql
ql
⎠ across l+ , (3.46b)
⎝
Ṽ1 (x, k) = −
λl + k
i(λl −λr )x
+
−2iλr x
e
r (k) e
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⎛
r̄ + (k) e2iλr x
iql ⎝
∗
Ṽ2 (x, k) = −
ql
+
−
λl + k
−
r̄
(k)r̄
(k)
ei(λr −λl )x
ql
ei(λr −λl )x
−
−r̄ (k) e
−2iλl x
⎞
⎠
across l− .
(3.46c)
As for the RH problem in terms of scattering coefficients from the right, the jump matrices
Ṽ1 (x, k) and Ṽ2 (x, k) satisfy the following upper/lower half plane symmetry:
Ṽ2 (x, k) = σ2 Ṽ1∗ (x, k ∗ )σ2 .
In the RH problem across r+ \ l+ , one has
+
M̃ (x, k) = φ + (x, k) eiλl x
M̃ (x, k) = φ − (x, k) eiλl x
−
ψ + (x, k) −iλr+ x
,
e
c+ (k)
ψ − (x, k) −iλr− x
.
e
c− (k)
Note, however, that unlike what happens in the RH problem from the right, here one cannot use
(3.19a)–(3.19d) to determine the jump. The same holds for the RH problem on r− \ l− . In fact, in
both Eqs. (3.19b) and (3.19d), the right-hand sides are only simultaneously defined for k ∈ R ∪ l ,
and cannot be extended on either r+ \ l+ or r− \ l− . This is also evident from (2.30c), where
it is clear that, unlike ρ ± (k) and ρ̄ ± (k), which can be respectively continued on r+ \ l+ and
r− \ l− , the reflection coefficients from the left, r ± (k) and r̄ ± (k), are only generically defined on
the continuous spectrum, i.e., for k ∈ R ∪ l .
In order to formulate the RH problem from the left on r \ l , one has to consider both pieces
of the cut r+ \ l+ and r− \ l− simultaneously, and take into account that: (i) λr changes sign
across r+ \ l+ and r− \ l− ; (ii) φ + (x, k) = φ − (x, k) for k ∈ r+ \ l+ , and φ̄ + (x, k) = φ̄ − (x, k)
for k ∈ r− \ l− ; (iii) ψ ± (x, k)/c± (k) and ψ̄ ± (x, k)/c̄± (x, k) are related to each other via the
symmetry relations (2.34), (2.38a) and (2.38b), i.e.,
ψ ± (x, k)
ψ̄ ∓ (x, k)
=
c± (x, k)
d̄ ∓ (x, k)
k ∈ r+ \ l+ ,
(3.47a)
ψ ∓ (x, k)
ψ̄ ± (x, k)
= ∓
±
c̄ (x, k)
d (x, k)
k ∈ r− \ l− .
(3.47b)
Solving the RH problem from the left (with poles, corresponding to the zeros of c(k) and c̄(k)
in the upper/lower half planes, which, by (2.28d) are the same as the ones from the right) amounts
to computing the sectionally meromorphic matrix M̃(x, k) with the given jumps Ṽ j , and normalized
to the identity as k → ∞. The potential is then reconstructed by the large k expansion of the latter,
since
i
M̃(x, k) = I2 +
Q(x)σ3 [1 + o(1)] .
2k
IV. TIME EVOLUTION OF THE SCATTERING DATA
According to the second of (2.1), the time-evolution of the eigenfunctions is given by
2ik 2 − i|q|2 −2kq − iqx
v,
vt =
2kq ∗ − iqx∗ −2ik 2 + i|q|2
(4.1)
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and asymptotically, taking into account q(x, t) → ql/r (t) = Al/r eiθl/r (t) as x → ∓∞,
⎛
⎞
2
2ik 2 − i Al/r
−2kql/r
⎠v
vt ⎝
as x → ∓∞ .
∗
2
2kql/r
−2ik 2 + i Al/r
(4.2)
The scattering problem in (2.1) as x → ∓∞ gives for the two components of any eigenfunction
v(x, t),
∗
v (1) −vx(2) + ikv (2) ,
ql/r
ql/r v (2) vx(1) + ikv (1) .
Introducing these equations into (4.2), we obtain as x → ∓∞,
2
v (1) − 2kvx(1) ,
vt(1) −i Al/r
2
vt(2) i Al/r
v (2) − 2kvx(2) .
(4.3)
The Jost solutions, whose boundary values as x → ∓∞ are given by (2.20a)–(2.20b), are not
compatible with the above time evolution. Therefore, we introduce time-dependent eigenfunctions
to be solutions of the evolution equation (4.1). For instance, let
ϕ(x, k, t) = ei A∞ t φ(x, k, t),
(4.4)
ϕt = i A∞ ϕ + ei A∞ t φt .
(4.5)
so that
Taking into account that the components of ϕ satisfy asymptotically the system (4.3) as x → −∞,
and that
1
0
−iλ
x
l
φ −iql∗ (t) e
, φt e−iλl x , x → −∞ ,
ql∗ (t)
−
θ̇
(t)
l
λl +k
λl +k
where dot denotes differentiation with respect to time, substituting into (4.5), the first component
yields
A∞ = 2kλl − Al2 ,
and from the second component we have: θ̇l (t) = −2Al2 , so that
θl (t) = −2Al2 t + θl (0) .
In a similar way, we can determine the evolution of the asymptotic phase as x → +∞: θr (t) =
−2Ar2 t + θr (0), as well as the time evolution of the other Jost solutions, obtaining for = (φ φ̄)
and = (ψ̄ ψ),
(4.6a)
∂t = i(2k 2 − |q|2 + Q x )σ3 − 2k Q − i(2kλl − Al2 ) σ3 ,
∂t = i(2k 2 − |q|2 + Q x )σ3 − 2k Q − i(2kλr − Ar2 ) σ3 .
(4.6b)
Differentiating (2.24a) with respect to t and taking into account the time evolution of the Jost
solutions (4.6a)–(4.6b), we obtain for the scattering matrix
∂t S = i(2kλr − Ar2 )σ3 S − i(2kλl − Al2 )S σ3 .
(4.7)
Explicitly, this yields the following expressions for the time evolution of the scattering coefficients
a(k, t) and b(k, t), and the reflection coefficient from the right ρ(k, t) = b(k, t)/a(k, t),
a(k, t) = a(k, 0) ei[2k(λr −λl )−Ar +Al ]t ,
(4.8a)
b(k, t) = b(k, 0) ei[−2k(λr +λl )+Ar +Al ]t ,
(4.8b)
ρ(k, t) = ρ(k, 0) e−2i(2kλr −Ar )t .
(4.8c)
2
2
2
2
2
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The first equation shows that the discrete eigenvalues kn are time independent, and given by the
zeros of a(k, 0). Note that in the symmetric case Al = Ar , and therefore a(k, t) = a(k, 0), i.e., the
transmission coefficient is time-independent. Moreover, for the large k behavior of a(k, t), taking
into account (A11), one still finds from (4.8a) that, consistently with (2.47a), a(k, t) ∼ 1 as |k| → ∞
for k ∈ Kr+ ∪ R and for all t ≥ 0; it ensures that the inverse problem is well posed. Similarly, we
can find the evolution of the other scattering coefficients c(k, t), d(k, t) etc., and of the reflection
coefficient from the left r (k, t) = d(k, t)/c(k, t),
r (k, t) = r (k, 0) e2i(2kλl −Al )t .
2
(4.9)
Finally, we need to determine the time dependence of the norming constants. Differentiating
φ(x, kn ) = bn ψ(x, kn ) with respect to time and evaluating the first column of (4.6a) and the second
column of (4.6b) at k = kn , we get
bn (t) = bn (0)e−i[2kn (λl (kn )+λr (kn ))−Al −Ar ]t ,
2
2
n = 1, . . . , N .
Then from the definition of the norming constants in (2.39a), we obtain
Cn (t) = Cn (0)e−2i[2kn λr (kn )−Ar ]t ,
2
n = 1, . . . , N .
(4.10)
V. CONCLUSIONS
We have developed the IST for the focusing NLS with fully asymmetric non-zero boundary
conditions as x → ±∞. This is a highly nontrivial generalization of the case where the amplitudes
of the background field are taken to be the same at both space infinities (see Ref. 12), and it
involves dealing with additional technical difficulties, the most important of which being the fact
that when the amplitudes of the NLS solutions as x → ±∞ are different, in the spectral domain
one cannot introduce a uniformization variable that allows mapping the multiply sheeted Riemann
surface for the scattering parameter to a single complex plane. Important differences with respect
to the symmetric case obviously also arise in the inverse problem, where, in addition to solitons
(corresponding to the discrete eigenvalues of the scattering problem), and to radiation (corresponding
to the continuous spectrum of the scattering operator, and represented in the inverse problem by
the reflection coefficients for k ∈ R ∪ (−i Al , i Al )), one also has a nontrivial contribution from the
transmission coefficients for k ∈ (−i Ar , −i Al ) ∪ (i Al , i Ar ), as shown by the last term is (3.27),
contributing to the left Marchenko equations. Correspondingly, (2.37a) and (2.37c) show that in the
right Marchenko equations one always has a nontrivial contribution from the integral terms in (3.12)
and (3.14), since ρ(k) [resp. ρ̄(k)] cannot vanish for k ∈ (i Al , i Ar ) [resp. k ∈ (−i Ar , −i Al )]. In
particular, this implies that no pure soliton solutions exist, and solitons are always accompanied by
a radiative contribution of some sort. As a consequence, unlike the symmetric case, here no explicit
solution can be obtained by simply reducing the inverse problem to a set of algebraic equations.
The results presented in this paper will pave the way for the investigation of the long-time
asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the
nonlinear steepest descent method, in analogy to what was done, for instance, in Ref. 17 for the
modified KdV equation, or in Refs. 13 and 14 for the focusing NLS with step-like initial conditions.
Moreover, the Marchenko integral equations obtained here will provide an alternative setup for the
study of the long-time behavior of the solutions by means of matched asymptotics, as was recently
done for KdV in Ref. 1.
The study of the long-time asymptotics, as well as the derivation of solutions describing solitons
superimposed to small radiation, will be the subject of future investigation.
ACKNOWLEDGMENTS
F.D. and F.V. wish to express their gratitude for the hospitality of the Department of Mathematics
of the University of Colorado Colorado Springs where part of the work was conducted. B.P. would
like to acknowledge Mark Ablowitz and Gino Biondini for many valuable discussions. This research
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is partially supported by NSF, under Grant Nos. DMS-1009248 and DMS-1311883, and by MIUR
under PRIN 2010-2011 “Geometric and analytic theory of Hamiltonian systems in finite and infinite
dimensions.”
APPENDIX: NORM ESTIMATES OF THE GROUPS e x l /r (k )
2
2
Taking into account that r/l
(k) = −λr/l
I2 , it is straightforward to obtain from its series
x r/l (k)
as
representation the explicit expression of e
e xr/l (k) = cos (λr/l x) I2 +
sin (λr/l x)
sin (λr/l x)
Q r/l − ik
σ3 .
λr/l
λr/l
(A1)
We recall that the Hilbert-Schmidt norm of a matrix A is defined as: A2HS = trace(A† A), while
the spectral norm is the square root of the largest singular value of A† A.Clearly, det e xr/l (k) = 1
for any k ∈ C, and for the Hilbert-Schmidt norm of the group for λr/l =
k ∈ R ∪ [−i Ar/l , i Ar/l ], one has
2
e xr/l (k) HS =: 2F(x, k) ,
F(x, k) ≡ cos2 (λr/l x) +
2
k 2 + Ar/l
∈ R, i.e., for
2
|k|2 + Ar/l
2
λr/l
sin2 (λr/l x) .
2
2
2
Note that F(x, k) ≡ 1 for k ∈ R [since in this case |k|2 + Ar/l
= k 2 + Ar/l
= λr/l
], and
2
2
F(x, ±i Ar/l ) ≡ 1 + 2Ar/l x , which is obtained from the above in the limit λr/l → 0. Also,
2
2
note that for k ∈ (−i Ar/l , i Ar/l ) it is |k|2 = −k 2 and λr/l
≤ Ar/l
. Hence for k ∈ (−i Ar/l , i Ar/l ),
one has
F(x, k) = cos (λr/l x) +
2
2
|k|2 + Ar/l
2
λr/l
sin (λr/l x) ≡ 1 +
2
2
2
− λr/l
)
2(Ar/l
2
λr/l
sin2 (λr/l x) ≥ 1 ,
(A2a)
and for k ∈ [−i Ar/l , i Ar/l ],
2
F(x, k) ≤ 1 + 2Ar/l
x2 .
(A2b)
The bounds (A2) obviously also hold for k ∈ R. Using the identity
2
2
1
4
2
A = 2 AHS + AHS − 4| det A|
for the squared spectral norm of a 2 × 2 matrix A, we then get
2
e xr/l (k) = F(x, k) + F 2 (x, k) − 1,
k ∈ R ∪ (−i Ar/l , i Ar/l ),
yielding
Cr/l (k) = sup e
xr/l (k)
1,
= x∈R
F(k) +
k ∈ R,
F 2 (k) − 1, k ∈ (−i Ar/l , i Ar/l ),
(A3)
where
F(k) = 1 +
2
2
− λr/l
)
2(Ar/l
2
λr/l
=1+
2|k|2
2
λr/l
k ∈ (−i Ar/l , i Ar/l ) .
Finally, for k ∈ [−i Ar/l , i Ar/l ] ∪ R and for each x ∈ R, from (A2b) we have
2
2
2
2
(1 + |x|)2 ,
e xr/l (±i Ar/l ) = 1 + 2Ar/l
x 2 + (1 + 2Ar/l
x 2 )2 − 1 ≤ C̃r/l
(A4)
where C̃r/l is a positive constant, independent of x ∈ R.
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Gronwall’s inequality. Consider the integral equation
∞
dy B(x, y)U (y),
U (x) = A(x) +
(A5)
x
where A(x) and B(x, y) are, respectively, vector-valued and matrix-valued continuous functions for
all x ∈ R, such that
A(x) ≤ α(x),
B(x, y) ≤ β(y) ,
(A6)
for some α(x), β(x) real-valued, continuous, non-negative functions; moreover, α(x) is assumed to
be non-increasing for all x ∈ R, and β(x) ∈ L 1 (a, ∞) for all a ∈ R. Then the Neumann series
U (x) =
∞
Un (x) ,
U0 (x) = A(x) ,
∞
Un+1 (x) =
dy B(x, y)Un (y) ,
(A7)
x
n=0
is uniformly convergent and it provides the (unique) solution of the integral equation (A5). Since the
Neumann series is a uniformly convergent series of continuous functions, U (x) is itself a continuous
function for all x ∈ R, and it satisfies Gronwall’s inequality
∞
dz β(z) .
(A8)
U (x) ≤ α(x) exp
x
The above results can be generalized in the obvious way to the case when the integral equation (A5)
and, consequently, the bounds (A6) contain a parametric dependence on k in A(x, k), B(x, y, k),
and U (x, k).
Proof of Gronwall’s inequality. The proof of the above results relies on the fact that from the
bounds (A6) it follows that for all non-negative integers n,
∞
∞
n
n
∞
∞
1
1
Un (x) ≤ α(x)
dz β(z)
⇒
Un (x) ≤ α(x)
dz β(z) .
n!
n!
x
x
n=0
n=0
Indeed, the above estimate is obviously true for n = 0. Assuming it to be true for a certain n, we
obtain
∞
dy B(x, y)Un (y)
Un+1 (x) ≤
x
n
∞
1
≤
dy α(y)β(y)
dz β(z)
n!
x
y
∞
n
∞
1
≤ α(x)
dy β(y)
dz β(z)
n!
x
y
&
∞
n+1 '∞
−1
= α(x)
dz β(z)
(n + 1)!
y
∞
= α(x)
1
(n + 1)!
∞
y=x
n+1
dz β(z)
,
x
as claimed. Note that we have used that α(x) is non-increasing.
Proof of Proposition 2.1. For k ∈ R ∪ (−i Ar , i Ar ), the estimate (A3) shows that e xr (k) ≤
Cr (k) for each x ∈ R. Then the result for (2.8a) follows by applying Gronwall’s inequality with
α(x, k) = Cr (k), β(y, k) = Cr (k)Q(y) − Q r , yielding
˜
(x,
k) ≤ Cr (k)eCr (k)
%∞
x
dy Q(y)−Q r .
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For k ∈ R ∪ [−i Ar , i Ar ], from the estimate (A4) it follows that there exists a constant C̃r such that
e xr (k) ≤ C̃r (1 + |x|) for each x ∈ R. By applying Gronwall’s inequality we then arrive at
˜
(x,
k) ≤ C̃r (1 + |x|)eC̃r
%∞
x
dy (1+|y−x|)Q(y)−Q r ≤ C̃r (x0 )(1 + |x|)eC̃r (x0 )
%∞
x
dy(1+|y|)Q(y)−Q r ,
where C̃r (x0 ) = C̃r (1 + max(0, −x0 )) and x ≥ x0 .
Proof of Proposition 2.2. Multiplying (2.8a) and (2.8b) from the right by the appropriate columns
of the matrices Wr/l (k), respectively, and using the explicit expressions (A1) for e(x−y)Ar/l (k) , we
obtain the following Volterra integral equations for the Jost solutions:
∞
dy r− (y − x, k)[Q(y) − Q r ]eiλr y ψ̄(y, k),
(A9a)
eiλr x ψ̄(x, k) = Wr,1 (k) −
x
e−iλr x ψ(x, k) = Wr,2 (k) −
∞
x
eiλl x φ(x, k) = Wl,1 (k) +
x
−∞
e−iλl x φ̄(x, k) = Wl,2 (k) +
dy r+ (y − x, k)[Q(y) − Q r ]e−iλr y ψ(y, k),
dy l+ (x − y, k)[Q(y) − Q l ]eiλl y φ(y, k),
x
−∞
(A9c)
dy l− (x − y, k)[Q(y) − Q l ]e−iλl y φ̄(y, k),
where the subscripts j = 1, 2 in the matrices Wl/r (k) denote their jth column, and
−2iλ x
iqr −2iλr x
r −k
r
1 + λ2λ
−1
− 2λ
−1
e
e
−
r
r
r (x, k) =
−2iλ x
,
iqr∗ −2iλr x
r −k
r
e
e
−1
e−2iλr x − λ2λ
−
1
2λr
r
r+ (x, k)
=
l+ (x, k)
=
l− (x, k)
=
λr −k
e2iλr x −
2λr
∗ iq
− 2λrr e2iλr x − 1
e2iλr x −
λl −k
e2iλl x −
2λl
∗ iql
e2iλl x − 1
2λl
1+
1
1
(A9d)
(A10a)
,
1
(A10b)
iql 2iλl x
e
− 2λ
−1
l
2iλ x
,
l −k
e l −1
e2iλl x − λ2λ
l
(A10c)
1
λl −k
e−2iλl x −
2λl
iql∗ −2iλl x
− 2λl e
−1
e−2iλl x −
iqr 2iλr x
e
−1
2λr
2iλ x
r −k
e r −
+ λ2λ
r
(A9b)
1
1
iql −2iλl x
e
−1
2λl
−2iλ x
l −k
l
e
+ λ2λ
−
l
.
1
(A10d)
With the given choice of the branch cuts, we easily derive the following expressions for the
behavior of λr/l as k → ∞:
A2 Ar2 1 + O(k −2 ) ,
λl − k = l 1 + O(k −2 ) .
2k
2k
Moreover, for x ≥ 0 we have
±2iλr x
e
− 1 x
±2iλr z ≡
dz e
k ∈ Kr± ∪ ∂Kr± ,
≤ min(x, 1/|λr |),
2λr
0
λr − k =
(A11)
(A12)
and similarly for the l-subscripted quantities. By the maximum modulus principle, we also get for
k ∈ Kr± ∪ ∂Kr± ,
&
2 '1/2
√
T 1
T
∗
(1 , −iqr /(λr + k)) = 1 , −i(λr − k)/q = 1 +
max± |λr − k|
= 2,
r
|qr | k∈∂Kr
(A13)
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where the spectral norm is used. Now applying (A11)–(A13) to estimate
−
(x, k) ≤ 1 + 2|qr | min(x, 1/|λr |),
k ∈ Kr− ∪ ∂Kr− ,
r
(A14)
we obtain with the help of Gronwall’s inequality
√ %∞
iλ x
e r ψ̄(x, k) ≤ 2 e x dy [1+2|qr |(y−x)]Q(y)−Qr ,
(A15)
where the estimate is uniform for (x, k) ∈ [x0 , +∞) × [Kr− ∪ ∂Kr− ∪ ∂Kr+ ] under the hypothesis
(H 1 ). Thus, under the hypothesis (H 1 ) and for each x ∈ R, the Jost solution ψ̄(x, k) is continuous
in k ∈ Kr− ∪ ∂Kr− ∪ Kr+ and analytic in k ∈ Kr− . Likewise we obtain estimates for the other three
Jost solutions, which proves the continuity and analyticity properties mentioned above.
Proof of Proposition 2.3. Differentiating (A9a) with respect to k we obtain the integral equation
∂ iλr x
∂ ∞
e ψ̄(x, k)− Wr,1 (k) = −
dy r− (y −x, k)[Q(y)− Q r ]Wr,1 (k)
∂k
∂k x
∞
(
)
∂ −
r (y − x, k) [Q(y) − Q r ] eiλr y ψ̄(y, k) − Wr,1 (k)
−
dy
∂k
x
∞
∂ iλr y
e ψ̄(y, k) − Wr,1 (k) ,
−
dy r− (y − x, k)[Q(y) − Q r ]
∂k
x
where, from (A10a),
k
∂ −
−2iλr (y−x) 0 0
(y − x, k) = − 2i (y − x)e
0 1
∂k r
λr
−2iλr (y−x)
e
k λr − k
−1
−iqr
−2iλr (y−x)
− 2
+ 2i(y − x)e
iqr∗
−(λr − k)
2λr
λr
−
λr − k (e−2iλr (y−x) − 1)
σ3 .
λr
2λr
Using (A12), we obtain
∂ −
(y − x, k) ≤ 2 |k| (y − x) 1 + |λr − k| + |qr | + |λr − k| (y − x) .
∂k r
|λr |
|λr |
|λr |
(A16)
A simple Gronwall argument then suffices to estimate its solution (for k = ±i Ar ). A similar result
holds for ψ(x, k), as well as for the other two Jost solutions, under the hypothesis k = ±i Al .
Proof of the inversion formulas (3.3). Consider the representation (3.1), which we can write as
∞
xr (k)
˜
(x, k) = e
+
dy K (x, y)e yr (k)
x
∞
sin(λr x)
sin(λr y)
Qr +
dy K (x, y) cos(λr y) +
Qr
λr
λr
x
∞
sin(λr x)
sin(λr y)
− ik
σ3 − ik
dy K (x, y)
σ3
λr
λr
x
= cos(λr x)I2 +
= cos(λr x)I2 +
− ik
where
sin(λr x)
Q r + c (x, λr ) + s (x, λr )Q r
λr
sin(λr x)
σ3 − ik s (x, λr )σ3 ,
λr
∞
c (x, λr ) =
x
dy K (x, y) cos(λr y) ,
∞
s (x, λr ) =
dy K (x, y)
x
sin(λr y)
.
λr
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Separating the parts that are even and odd (in λr ), we obtain
˜
˜
(x,
k) + (x,
−k)
sin(λr x)
= cos(λr x)I2 +
Q r + c (x, λr ) + c (x, λr )Q r ,
2
λr
˜
˜
sin(λr x)
(x,
k) − (x,
−k)
= −ik
σ3 − ik s (x, λr )σ3 ,
2
λr
and as a result,
˜
˜
(x,
k) − (x,
−k)
sin(λr x)
σ3 −
I2 ,
−2ik
λr
˜
˜
˜
˜
(x,
k) + (x,
−k)
(x,
k) − (x,
−k)
− cos(λr x)I2 +
σ3 Q r .
c (x, λr ) =
2
2ik
s (x, λr ) =
If we then depart from the identity
∞
dy K (x, y)eiλr y = c (x, λr ) + iλr s (x, λr ) ,
x
we can write
∞
dy K (x, y)eiλr y =
x
−
˜
˜
˜
˜
(x,
k) + (x,
−k)
(x,
k) − (x,
−k)
− cos(λr x)I2 +
σ3 Q r
2
2ik
λr ˜
˜
[(x, k) − (x,
−k)]σ3 − i sin(λr x)I2 ,
2k
i.e.,
i
λr
˜
dy K (x, y)eiλr y = 12 (x,
k) I2 − σ3 Q r − σ3
k
k
x
i
λr
˜
+ 12 (x,
−k) I2 + σ3 Q r + σ3 − eiλr x I2 .
k
k
∞
(A17)
Now, the following identities can be easily verified:
i
λr
iλr x
I2 − σ3 Q r − σ3 = e
I2 −
e
k
k
i
λr
e xr (−k) I2 + σ3 Q r + σ3 = eiλr x I2 +
k
k
xr (k)
i
λr
σ3 Q r − σ3 ,
k
k
i
λr
σ3 Q r + σ3 .
k
k
Multiplying either side of (A17) by e−iλr x , and using the above identities, we find
∞
dy K (x, y)e
x
iλr (y−x)
i
λr
I2 − σ3 Q r − σ3
=
k
k
i
λr
˜
+ 12 (x,
−k)e−xr (−k) I2 + σ3 Q r + σ3 − I2
k
k
i
λr
˜
k)e−xr (k) − I2 I2 − σ3 Q r − σ3
= 12 (x,
k
k
i
λr
−xr (−k)
1 ˜
+ 2 (x, −k)e
− I2 I2 + σ3 Q r + σ3 .
k
k
1 ˜
(x, k)e−xr (k)
2
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Consequently, since
y < x, we arrive at
%∞
x
J. Math. Phys. 55, 101505 (2014)
dy K (x, y)eiλr (y−x) ≡
%∞
−∞
dy K (x, y)eiλr (y−x) because K (x, y) ≡ 0 for
i
λr
˜
(x,
k)e−xr (k) − I2 I2 − σ3 Q r − σ3
k
k
−∞
i
λr
˜
.
+ (x,
−k)e−xr (−k) − I2 I2 + σ3 Q r + σ3
k
k
K (x, y) =
1
4π
∞
dλr e−iλr (y−x)
Proof of Proposition 3.1. Substituting (3.2a) into (2.1) and multiplying the resulting equation
from the right by eiλr σ3 x , we get
∂x (x, k)eiλr σ3 x − iλr (x, k)σ3 eiλr σ3 x
∞
−iλr σ3 (s−x)
.
= (−ikσ3 + Q(x)) Wr (k) +
ds K (x, s)Wr (k)e
x
By using (2.18), (2.45a)–(2.45b), (2.46) (where in (2.46) we assume that ∂x q ∈ L 1 (R)), we obtain
i Q(x)σ3
I2 +
σ3 [1 + o(1)]
2k
∞
−iλr σ3 (s−x)
= (−ikσ3 + Q(x)) I2 +
ds K (x, s)e
i∂x Q(x)σ3
− iλr
2k
x
+
i
(−ikσ3 + Q(x)) Q r +
λr + k
∞
ds K (x, s)Q r e−iλr σ3 (s−x) σ3 ,
x
i.e.,
ikσ3 + 12 Q(x) + o(1/k)
∞
= −ikσ3 + Q(x) + Q(x)
∞
− iλr σ3
ds K (x, s)e−iλr σ3 (s−x)
x
ds K (x, s)e−iλr σ3 (s−x) +
x
i Ar2
σ3
λr + k
∞
ds K (x, s)e−iλr σ3 (s−x)
x
Ar2
i
Q(x)Q r σ3
σ3 Q r σ3 +
+ 12 1 −
2
(λr + k)
λr + k
∞
i
(−ikσ3 + Q(x))
+
ds K (x, s)Q r e−iλr σ3 (s−x) σ3 .
λr + k
x
Now on the right-hand side, the third, fifth, and last terms involve Fourier integrals of matrix functions
with entries in L 2 (R; dλr ), multiplied by factors which are at least bounded at large k; therefore all
such terms vanish as k, λr → ∞. Next, we write for the fourth term
∞
− iλr σ3
ds K (x, s)e−iλr σ3 (s−x) =
x
∞
− iλr σ3
x
ds K (x, s)Wr (k)e−iλr σ3 (s−x) −
λr
σ3
λr + k
∞
ds K (x, s)Q r σ3 e−iλr σ3 (s−x) ,
x
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where the second term on the right has its entries in L 2 (R; dλr ). The first term on the right is written
as
∞
− iλr σ3
ds K (x, s)Wr (k)e−iλr σ3 (s−x)
x
= σ3 −iλr
∞
ds K (x, s)Wr (k)e−iλr σ3 (s−x) σ3 σ3
x
= σ3 −K (x, x)Wr (k) +
∞
ds [∂x K (x, s)] Wr (k)e−iλr σ3 (s−x)
x
∞
− ∂x
ds K (x, s)Wr (k)e
−iλr σ3 (s−x)
σ3
x
= σ3 −K (x, x)Wr (k) +
i
+
λr + k
∞
ds [∂x K (x, s)] e−iλr σ3 (s−x)
x
∞
ds [∂x K (x, s)] Q r σ3 e
−iλr σ3 (s−x)
iλr σ3 x
− ∂x (x, k)e
− Wr (k) σ3 .
x
Now let us examine the various terms within brackets on right-hand side of this last identity. The first
term is simply −K (x, x) + o(1). According to (2.46), the last term vanishes as k → ±∞, the second
term has its entries in L 2 (R; dλr ) (being the Fourier transform of an L 2 matrix function), and the
third term of the last member is L 2 multiplied by a bounded factor. Thus, dropping all contributions
that vanish as k → ±∞, and using that σ3 Q r σ3 = −Q r , we obtain
−ikσ3 + 12 Q(x) = −ikσ3 + Q(x) − σ3 K (x, x)σ3 − 12 Q r ,
i.e., the first of (3.4). The second equality in (3.4) can be proved in a similar way.
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